1 Introduction↩︎

1.1 Toda conformal field theory↩︎

Two-dimensional conformal field theory (CFT in short) provides a universal language for describing critical phenomena in statistical physics. At a second-order phase transition, a lattice model is expected to lose its microscopic length scale and, after suitable rescaling, to converge to a continuum theory whose main quantities of interest, the correlation functions, are invariant under conformal transformations.

This principle is especially powerful in two dimensions: the local conformal symmetry is infinite-dimensional, and its infinitesimal symmetries are encoded by the Virasoro algebra. The seminal work of Belavin, Polyakov and Zamolodchikov [2] showed how to exploit this symmetry to obtain strong constraints on correlation functions. Such constraints lead to the conformal bootstrap, a recursive procedure which aims to determine higher-point correlation functions from lower-point data, in particular from three-point functions, also known as the structure constants.

Liouville theory is a central example where this bootstrap philosophy can be carried out explicitly. Its origin goes back to Polyakov’s path-integral approach to two-dimensional quantum geometry and non-critical string theory [3]. The resulting theory is an interacting CFT whose structure constants, described by the DOZZ formula, were later independently predicted by Dorn-Otto [4] and Zamolodchikov-Zamolodchikov [5]. Together with the conformal block decomposition, these structure constants determine higher-point correlation functions on the sphere, or more generally on any Riemann surface via Segal’s gluing [6], and provide one of the basic non-rational examples of the conformal bootstrap.

In many models, Virasoro symmetry is only part of the structure. Certain critical systems possess additional conserved quantities, leading to extended chiral symmetry algebras and higher-spin symmetries. Among the most important examples are \(W\)-algebras, introduced by Zamolodchikov [7], which extend the Virasoro algebra by currents of higher conformal spin; see also [8] for a review. Toda CFTs provide a natural family of CFTs governed by such \(W\)-symmetries: the Virasoro symmetry of Liouville theory is replaced by the \(W\)-algebra associated with the underlying simple Lie algebra. From the point of view of statistical physics, such extended symmetries are expected to describe continuum limits of models with richer algebraic or geometric features, for instance, \(q\)-state Potts model and web models [9], [10]. This provides one motivation for studying Toda CFTs.

1.2 Compactified imaginary Toda CFT↩︎

The bootstrap approach gives a powerful way to characterize conformal field theories from their symmetry constraints, but it does not by itself provide a direct construction of the path integral. A complementary point of view is the probabilistic construction of CFTs, where the formal path integral is interpreted as an expectation over random fields. In Liouville theory, this approach is based on the Gaussian free field (see [11], [12]) and the Gaussian multiplicative chaos (see [13]), and has led to a rigorous construction of correlation functions and their conformal covariance properties [14] and also a rigorous proof of the DOZZ formula [15], thereby connecting the probabilistic and bootstrap perspectives. Moreover, the gluing axiom and the bootstrap strategy have also been established [16], [17].

More recently, the same probabilistic perspective has also been developed for Toda CFTs, leading to a rigorous construction of Toda correlation functions as well as to the study of their conformal covariance and \(W\)-symmetry constraints [18], [19]. On the bootstrap side, Toda structure constants are expected to be governed by higher-rank analogs of the DOZZ formula. In particular, Fateev and Litvinov proposed explicit formulae for certain Toda structure constants [20], which have been proved under the probabilistic framework in the case \(\mathfrak{g}=\mathfrak{sl}_3\) [21]. These results provide the higher-rank background for the compactified imaginary Toda theory considered in this paper.

The goal of this paper is to develop an analogous construction for a compactified imaginary version of Toda theory. Our construction is inspired by, and follows closely, the framework developed in [1]. Let \((\Sigma,g)\) be a Riemann surface, let \(\mathfrak g\) be a complex simple Lie algebra of rank \(r\), with simple roots \(e_1,\ldots,e_r\), and let \(\mathfrak a\) be the real Cartan subspace. For \(\gamma>0\), we consider fields \(\Phi:\Sigma\rightarrow\mathbb{T}(\gamma)\) taking values in the torus \[\mathbb{T}(\gamma)=\mathfrak a/(2\pi\Lambda), \qquad \Lambda=\gamma^{-1}\bigoplus_{i=1}^r\mathbb{Z}\omega_i^\vee .\] Formally, the compactified imaginary Toda action is\[\label{eq:Toda32equation} S_{\mathfrak g}(\Phi,g)=\frac{1}{4\pi}\int_\Sigma\left(\langle \partial_g\Phi,\partial_g\Phi\rangle_g+K_g\langle \boldsymbol{\mathrm{i}}Q,\Phi\rangle+ 4\pi\sum_{i=1}^r\mu_i e^{\boldsymbol{\mathrm{i}}\gamma\langle e_i,\Phi\rangle} \right)\mathrm{d}\mathrm{v}_g,\tag{1}\] where \(\boldsymbol{\mathrm{i}}:=\sqrt{-1}\), and \(Q=\gamma\rho-\frac{2}{\gamma}\rho^\vee\) is the background charge with \(\rho\) the Weyl vector and \(\rho^\vee\) its dual. For a functional \(F\) of the field \(\Phi\), we define \[\langle F\rangle_{\Sigma,g}:=\int_{\Phi:\Sigma\rightarrow\mathbb{T}(\gamma)}F(\Phi)e^{-S_\mathfrak{g}(\Phi,g) }\mathrm D\Phi,\] where \(\mathrm{D}\Phi\) is formally the Lebesgue measure on the function space \(\{\Phi:\Sigma\rightarrow\mathbb{T}(\gamma)\}.\)

The compactification makes the interaction terms \(e^{i\gamma\langle e_i,\Phi\rangle}\) globally well defined as characters of \(\mathbb{T}(\gamma)\), but it also forces the theory to include topological sectors. More precisely, a smooth map \(\Phi:\Sigma\rightarrow\mathbb{T}(\gamma)\) induces an \(\mathfrak{a}\)-valued closed \(1\)-form \(\Omega\) with periords in \(2\pi\Lambda\), and the Hodge decomposition allows one to uniquely decompose the form as \(\Omega=\Omega_{\mathrm{h}}+\mathrm{d}f,\) with \(\Omega_{\mathrm{h}}\) a harmonic \(\mathfrak{a}\)-valued 1-form and \(f\) a smooth \(\mathfrak{a}\)-valued function. This shows that any smooth \(\mathbb{T}(\gamma)\)-valued map \(\Phi\) admits an orthogonal decomposition: \(\Phi=f+I_{x_0}(\Omega_{\mathrm{h}}),\) where \(I_{x_0}(\Omega_\mathrm{h}):=\int_{\gamma_{x_0,x}}\Omega_{\mathrm{h}}\) is a multivalued harmonic function, with \(\gamma_{x_0,x}\) a path from \(x_0\) to \(x\), and the orthogonality reads \[\int_\Sigma |\mathrm{d}\Phi|_g^2\mathrm{d}\mathrm{v}_g=\int_\Sigma|\mathrm{d}f|_g^2\mathrm{d}\mathrm{v}_g+\int_\Sigma \Omega_{\mathrm{h}}\wedge*\Omega_{\mathrm{h}}.\] Thus, formally, we can rewrite the path integral with both \(Q\) and \(\boldsymbol{\mu}\) vanishing as \[\begin{align}\langle F\rangle_{\Sigma,g}=&\int_{\Phi:\Sigma\rightarrow\mathbb{T}(\gamma)}F(\Phi)e^{-\frac{1}{4\pi}\int_\Sigma|\mathrm{d}\Phi|_g^2\mathrm{d}\mathrm{v}_g}\mathrm{D}\Phi\\ =&\int F(f+I_{x_0}(\Omega_{\mathrm{h}})+c)e^{-\frac{1}{4\pi}\int_\Sigma (|d f|^2_g+\Omega_{\mathrm{h}}\wedge*\Omega_{\mathrm{h}})\mathrm{d}\mathrm{v}_g}\mathrm{D}f\mathrm{d}\mu(\Omega_{\mathrm{h}})\mathrm{d}c ,\end{align}\] where \(\mathrm{D}f\) is a formal Lebesgue measure over zero-mean \(\mathfrak{a}\)-valued functions, \(\mathrm{d}\mu(\Omega_\mathrm{h})\) is the counting measure on the \(\mathfrak{a}\)-valued de Rham cohomology group, and \(\mathrm{d}c\) is the Lebesgue measure on the torus \(\mathbb{T}(\gamma).\) We note that, due to the multivaluedness of the primitive \(I_{x_0}(\Omega_\mathrm{h})\), the functional \(F\) has to satisfy certain periodic conditions so that the path integral makes sense.

When the background charge \(Q\) is nonzero, one has to regularize the curvature term \[\int_\Sigma K_g\langle \boldsymbol{\mathrm{i}}Q,\Phi\rangle\mathrm{d}\mathrm{v}_g,\] to deal with the multivalued primitive. Following the strategy of [1], we do so by choosing cuts and adding explicit geodesic-curvature counterterms, in such a way that the resulting path integral is independent of the auxiliary choices up to the expected lattice ambiguities (see Section 4.1).

Let us now describe more explicitly the objects constructed in this paper. The basic observables are correlation functions of electric, magnetic, and electro-magnetic operators. An electric operator with charge \(\alpha\) is obtained by Wick-renormalizing the exponential \(e^{\boldsymbol{\mathrm{i}}\langle \alpha,\Phi(x)\rangle}\) at an insertion point \(x\in\Sigma\). A magnetic operator, on the other hand, inserts a prescribed monodromy \(m\in\Lambda\) around a marked point, and is implemented by a singular closed \(1\)-form \(\nu_{\mathbf{z},\mathbf{m}}\). Electro-magnetic operators \(V^g_{(\alpha,m)}(z,v)\) combine these two effects. When \(m\neq0\), they depend on a choice of tangent direction \(v\) at the insertion point and carry a non-trivial spin. We prove that the corresponding correlation functions are well defined under natural Seiberg-type assumptions and satisfy the expected conformal anomaly, diffeomorphism covariance, and spin covariance.

The three-point functions on the Riemann sphere, or structure constants, are expected to play the role of basic building blocks in the conformal bootstrap. In the compactified imaginary Toda setting, their computation leads to higher-rank Coulomb gas integrals of Dotsenko–Fateev type, with screening variables indexed by the simple roots of \(\mathfrak g\). In special cases, these integrals are closely related to the imaginary Fateev–Litvinov formula [22] for Toda structure constants. Thus the correlation functions constructed here provide a probabilistic realization of the Coulomb gas picture in the compactified imaginary Toda setting.

Finally, we establish Segal’s gluing axioms [6] for the theory. To do so, we construct the path integral on surfaces with analytic boundary and view the resulting boundary amplitudes as vectors in the Hilbert space associated with the boundary field. We then prove that these amplitudes compose under gluing of surfaces. The proof uses the Markov property of the Gaussian free field, but also requires bookkeeping for the compactified topological sectors, magnetic backgrounds, and regularized curvature terms. These features are specific to the compactified theory and are essential for obtaining a consistent CFT on surfaces of arbitrary topology.

We summarize the above results in Theorem 1 and refer to Theorem 10 and Proposition 14 for precise statements.

We make some remarks on the proof:

1.3 Future directions↩︎

We conclude by describing several directions which naturally follow from the present construction.

• Hamiltonian spectrum and conformal bootstrap. The Segal gluing identities proved in Section 6 suggest a natural continuation: to study the spectrum of the Hamiltonian, namely the generator of the semigroup of annuli, as in [16]. In the compactified theory, this spectral analysis should reflect the decomposition of the boundary field into its compact zero mode, winding sector, and oscillator modes. It is also naturally connected with the expected \(W\)-symmetry of the theory: one can use Cerclé’s probabilistic construction of \(W\)-algebras [26] to study the action of \(W\)-currents on the boundary Hilbert space \(\mathcal{H}_{\mathfrak{g},\gamma}\) and their compatibility with the amplitudes constructed here. Together, the spectral analysis of the Hamiltonian and the \(W\)-algebraic structure of the boundary theory form part of the analytic and representation-theoretic input needed for a conformal bootstrap approach to compactified imaginary Toda theory.

• Scaling limits of higher-rank lattice models. A major motivation for this work is its expected relation with scaling limits of lattice models with extended symmetries, especially web models. In such models [9], [10], one expects observables to be governed not only by Virasoro symmetry but by higher-rank \(W\)-symmetries. This suggests that the correlation functions and amplitudes constructed here should appear as continuum limits of suitable lattice observables.

  In rank one, i.e., when \(\mathfrak{g}=\mathfrak{sl}_2\), this has been achieved via conformal loop ensembles (CLE) [27], [28], which serve as a candidate for scaling limits of critical loop models. More precisely, in [29], using techniques from Liouville quantum gravity [30], [31], the multi-points connectivity of CLE are expressed in terms of the imaginary DOZZ formula. For Toda theories, the corresponding random geometry should be richer than a collection of loops. For example, it is expected that the full interface of the critical 3-states Potts model enjoys \(W_3\)-symmetry. In particular, the interfaces are not characterized by simply conformal invariance and Markov property. One can employ the Edwards–Sokal coupling in the continuum, established in [32], to obtain the desired interface of critical Potts models, but to describe the interface cleanly requires non-trivial effort.

1.3.0.1 Organization of the paper.

Section 2 collects the geometric, analytic, and Lie-theoretic preliminaries used throughout the paper. In Section 3, we recall the \(\mathfrak a\)-valued Gaussian free field and the imaginary Gaussian multiplicative chaos, along with their properties. Section 4 is devoted to the construction of the path integral and of the electric, magnetic, and electro-magnetic correlation functions on closed surfaces, together with their conformal covariance properties. In Section 5, we specialize to the Riemann sphere, derive the Dotsenko–Fateev type integral representation of the correlation functions, and prove the closed formula for the semidegenerate three-point structure constant in type \(A\). Section 6 proves Segal’s gluing axioms by constructing boundary amplitudes and establishing their compatibility under gluing. Finally, the appendix contains the complex Dotsenko–Fateev integral identity and the meromorphic-continuation argument used in the proof of the structure constant formula.

1.3.0.2 Acknowledgements.

The author thanks Baptiste Cerclé for proposing this problem and for his support throughout this work.

2 Preliminary background and notations↩︎

2.1 Lie-theoretic conventions↩︎

Let \(\mathfrak{g}\) be a complex simple Lie algebra of rank \(r\) and \(\mathfrak{h}\) be the Cartan subalgebra. Let \(\mathfrak{a}\cong (\mathbb{R}^r,\langle\cdot,\cdot\rangle)\) be a Euclidean space such that \(\mathfrak{h}^*\cong\mathfrak{a}\oplus \boldsymbol{\mathrm{i}}\mathfrak{a}\). We consider the simple roots \((e_i)_{i=1,\ldots,r}\) of \(\mathfrak{g}\) which is a basis of \(\mathfrak{a}\) and satisfies \[2\frac{\langle e_i,e_j\rangle}{\langle e_i,e_i\rangle}=A_{ij},\] where \(A\) is the Cartan matrix of \(\mathfrak{g}\). Following physics convention, we normalize the inner product \(\langle\cdot,\cdot\rangle\) so that the longest root has norm \(2\). The renormalizing constant is given by \(2h^\vee\), where \(h^\vee\) is called the dual Coxeter number.

We further define by \[\omega_i=\sum^r_{j=1}(A^{-1})_{ij}e_j,\] the basis \((\omega_i)_{i=1,\ldots,r}\) of \(\mathfrak{a}^*\) (identified with \(\mathfrak{a}\)) dual to \((e^\vee_i)_{i=1,\ldots,r}\) so that \(\langle e_i^\vee,\omega_j\rangle=\delta_{ij}\) where \(e_i^\vee:=2\frac{e_i}{\langle e_i,e_i\rangle}\) is the coroot. Let \(\rho=\sum^r_{i=1}\omega_i\) denote the Weyl vector and its dual \(\rho^\vee=\sum^r_{i=1}\omega^\vee_i\) with \(\langle\omega^\vee_i,e_j\rangle=\delta_{ij}\). The Weyl vector has norm [33] \[|\rho|^2=\frac{h^\vee\dim\mathfrak{g}}{12}.\]

The Toda field will take values in the torus \(\mathbb{T}=\mathbb{T}(\gamma):=\mathfrak{a}/(2\pi\Lambda)\) where \[\Lambda=\Lambda_\gamma:=\gamma^{-1}\bigoplus_{i=1}^r\mathbb{Z}\omega_i^\vee\] is the scaled coweight lattice. We equip \(\mathbb{T}(\gamma)=\mathfrak{a}/(2\pi\Lambda)\) with the Haar measure \(\mathrm{d}c\) induced by the Lebesgue measure on \(\mathfrak{a}\). Thus \[\int_{\mathbb{T}(\gamma)}\mathrm{d}c = \operatorname{Vol}(\mathbb{T}(\gamma)) = \left(\frac{2\pi}{\gamma}\right)^r\left(\det(A)\prod^r_{i=1}\frac{\langle e_i,e_i\rangle}{2}\right)^{-1/2}.\]

2.2 Riemann surfaces, parametrized boundaries, gluing, and cutting↩︎

We briefly recall the geometric setup in [1].

2.2.0.1 Closed surfaces

We first fix the geometric conventions used throughout the paper. A closed Riemann surface \((\Sigma,[g])\) will mean a compact connected oriented smooth surface \(\Sigma\), without boundary, endowed with a conformal class of Riemannian metrics \([g]:=\{e^\rho:\rho\in C^\infty(\Sigma)\}\). Equivalently, the conformal class determines a complex structure \(J\in C^\infty(\Sigma;\mathrm{End}(T\Sigma))\), \(J^2=-\mathrm{Id}\). In local holomorphic coordinates \(z=x+\boldsymbol{\mathrm{i}}y\), any compatible metric is of the form \(g=e^\rho |\mathrm{d}z|^2,\) and the Hodge star on one-forms is characterized by \(*\mathrm{d}x=\mathrm{d}y,\,*\mathrm{d}y=\mathrm{d}x.\) We denote by \(K_g\) the scalar curvature, by \(\mathrm{d}\mathrm{v}_g\) the Riemannian volume form, and by \(\Delta_g=\mathrm{d}^*\mathrm{d}\) the non-negative Laplace–Beltrami operator. The Gauss–Bonnet formula reads \[\label{eq:Gauss-Bonnet-closed} \int_\Sigma K_g\,\mathrm{d}\mathrm{v}_g = 4\pi\chi(\Sigma), \qquad \chi(\Sigma)=2-2\mathbb{g}\tag{3}\] with \(\mathbb{g}\) the genus of \(\Sigma.\) Moreover, if \(\widehat g=e^\rho g\), \(\rho\in C^\infty(\Sigma)\), then \[\label{eq:curvature-conformal-change} K_{\widehat g} = e^{-\rho}\left(\Delta_g\rho+K_g\right).\tag{4}\]

2.2.0.2 Surfaces with analytic parametrized boundary.

Let \(\mathbb{T}:=\{e^{\boldsymbol{\mathrm{i}}\theta}:\theta\in\mathbb{R}/2\pi\mathbb{Z}\}\) be the unit circle. A compact Riemann surface with analytic parametrized boundary is a compact oriented smooth surface \(\Sigma\) with boundary \(\partial\Sigma=\bigsqcup_{j=1}^{\mathbb{b}}\partial_j\Sigma,\) together with an atlas \((U_j,\omega_j)\) and smooth diffeomorphisms \(\zeta_j:\mathbb{T}\longrightarrow \partial_j\Sigma, \, j=1,\ldots,\mathbb{b},\) such that the following hold. The collection \(U_j\) is an open cover \(\Sigma\), and \(U_j\cap\partial_j\Sigma\neq\emptyset\) if and only if \(j\in[1,\mathbb{b}]\). There exists \(\delta\in(0,1)\) such that, for \(j\in[1,\mathbb{b}]\), \[\omega_j(U_j)=\mathbb{A}_\delta := \{z\in\mathbb{C}:\delta<|z|\le 1\}, \qquad \omega_j(\partial_j\Sigma)=\mathbb{T}.\] The transition maps are holomorphic in the interior, and \(\omega_j\circ\zeta_j:\mathbb{T}\to\mathbb{T}\) is real analytic, i.e. extends holomorphically to a neighborhood of \(\mathbb{T}\).

The above charts define a complex structure on \(\Sigma\). A Riemannian metric \(g\) is compatible with this complex structure if, in every chart, \((\omega_j^{-1})^*g=e^{\rho_j}|\mathrm{d}z|^2\) for some smooth real-valued function \(\rho_j\). The orientation of \(\Sigma\) induces an orientation on each boundary component. We say that \(\partial_j\Sigma\) is outgoing if the orientation induced by \(\zeta_j(e^{\boldsymbol{\mathrm{i}}\theta})\) agrees with the boundary orientation, and incoming otherwise. We set \[\varsigma_j= \begin{cases} -1, & \partial_j\Sigma \text{ is outgoing},\\ +1, & \partial_j\Sigma \text{ is incoming}. \end{cases}\] By replacing a boundary chart by its inverse coordinate if necessary, we may assume that the parametrizations are adapted to the signs: \[\zeta_j(e^{i\theta}) = \begin{cases} \omega_j^{-1}(e^{\boldsymbol{\mathrm{i}}\theta}), & \varsigma_j=-1,\\ \omega_j^{-1}(e^{-\boldsymbol{\mathrm{i}}\theta}), & \varsigma_j=+1. \end{cases}\]

An admissible metric is a compatible metric satisfying \[(\omega_j^{-1})^*g=\frac{|\mathrm{d}z|^2}{|z|^2} \qquad\text{on }\omega_j(U_j),\qquad j=1,\ldots,\mathbb{b}.\] Then every boundary component is geodesic, has length \(2\pi\), and \(K_g=0\) near \(\partial\Sigma\). We denote by \(\nu\) the inward-pointing unit normal vector field and by \(\mathrm{d}\ell_g\) the induced boundary measure.

2.2.0.3 Gluing and cutting

We now recall the gluing convention. Suppose that two parametrized boundary components \(\partial_j\Sigma\) and \(\partial_k\Sigma\) have opposite signs, one outgoing and one incoming. The glued surface is obtained by identifying \[\zeta_j(e^{\boldsymbol{\mathrm{i}}\theta})\sim \zeta_k(e^{\boldsymbol{\mathrm{i}}\theta}), \qquad \theta\in\mathbb{R}/2\pi\mathbb{Z}.\] Near the glued circle, the complex coordinate is obtained by using \(\omega_j\) on one side and \(1/\omega_k\) on the other side. Hence the result is again a Riemann surface. If the original metric is admissible, the two metrics glue to give a smooth compatible metric on the glued surface. The remaining boundary components keep their original parametrizations.

Conversely, if \(\mathcal{C}\subset\Sigma^\circ\) is an analytically embedded simple closed curve and if a neighborhood of \(\mathcal{C}\) is identified holomorphically with an annulus \(\{z:\delta<|z|<\delta^{-1}\}\), with \(\mathcal{C}\) corresponding to \(\mathbb{T}\), then cutting \(\Sigma\) along \(\mathcal{C}\) produces a bordered Riemann surface \(\Sigma_{\mathcal{C}}:=\Sigma\setminus\mathcal{C}\) completed by two new analytic boundary components. One of them is outgoing and the other is incoming.

2.3 Regularized determinant of the Laplacian and Green’s function↩︎

Let \((\Sigma,g)\) be a connected compact oriented Riemann surface, with or without boundary. We denote by \(\Delta_g=\mathrm{d}^*\mathrm{d}\) the non-negative Laplace–Beltrami operator. When \(\partial\Sigma=\emptyset\), we consider \(\Delta_g\) acting on functions on \(\Sigma\). When \(\partial\Sigma\neq\emptyset\), we consider the Dirichlet Laplacian, namely \(\Delta_g\) acting on functions vanishing on \(\partial\Sigma\).

The spectrum of \(\Delta_g\) is discrete: \[\mathrm{Sp}(\Delta_g)=\{\lambda_j\}_{j\in\mathcal{J}}, \qquad 0\le \lambda_0\le \lambda_1\le \lambda_2\le\cdots, \qquad \lambda_j\to+\infty,\] where \(\mathcal{J}=\mathbb{N}_0\) (resp. \(\mathcal{J}=\mathbb{N}\)) if \(\partial\Sigma=\emptyset\) (resp. if \(\partial\Sigma\neq\emptyset\)). The regularized determinant of \(\Delta_g\) is defined by zeta regularization. For \(\mathrm{Re}(s)\gg1\), we set \(\zeta_g(s):=\sum_{\lambda_j>0}\lambda_j^{-s}.\) This series converges for \(\mathrm{Re}(s)\gg1\) and admits a meromorphic continuation to \(\mathbb{C}\), which is holomorphic at \(s=0\). We then define \(\det{}'(\Delta_g):=\exp\bigl(-\zeta'_g(0)\bigr).\) If \(\partial\Sigma=\emptyset\) and \(\hat{g}=e^\rho g\) for some \(\rho\in C^\infty(\Sigma)\), then the determinant satisfies the Polyakov formula [34] \[\label{eq:Polyakov-formula} \log \frac{\det{}'(\Delta_{\hat{g}})}{\mathrm{v}_{\hat{g}}(\Sigma)} = \log \frac{\det{}'(\Delta_g)}{\mathrm{v}_g(\Sigma)} -\frac{1}{48\pi}\int_\Sigma\bigl(|d\rho|_g^2+2K_g\rho\bigr)\,\mathrm{d}\mathrm{v}_g.\tag{5}\]

We now recall the Green function of the Laplacian. If \(\partial\Sigma=\emptyset\), let \(\Pi_0:L^2(\Sigma,\mathrm{d}\mathrm{v}_g)\to \ker\Delta_g\) be the orthogonal projection onto the constants. The resolvent operator \(R_g:L^2(\Sigma,\mathrm{d}\mathrm{v}_g)\to L^2(\Sigma,\mathrm{d}\mathrm{v}_g)\) is defined by \[\Delta_gR_g=2\pi(\mathrm{Id}-\Pi_0), \qquad R_g^*=R_g, \qquad R_g1=0.\] Its integral kernel is the Green function \(G_g\), characterized by \[R_gf(x)=\int_\Sigma G_g(x,y)f(y)\,\mathrm{d}\mathrm{v}_g(y), \qquad f\in L^2(\Sigma,\mathrm{d}\mathrm{v}_g).\] Equivalently, \(G_g\) is the unique symmetric function on \(\Sigma\times\Sigma\setminus \{(x,x):x\in\Sigma\}\) such that, for each fixed \(x\in\Sigma\), \[\label{eq:Green-function-equations-closed} -\Delta_g G_g(x,\cdot)=2\pi\left(\delta_x-\frac{1}{\mathrm{v}_g(\Sigma)}\right), \qquad \int_\Sigma G_g(x,y)\,\mathrm{d}\mathrm{v}_g(y)=0.\tag{6}\]

If \(\partial\Sigma\neq\emptyset\), we denote by \(R_{g,D}\) the inverse of the Dirichlet Laplacian: \(R_{g,D}:=(2\pi)\Delta_g^{-1}.\) Its integral kernel is the Dirichlet Green function \(G_{g,D}\), characterized by \[R_{g,D}f(x)=\int_\Sigma G_{g,D}(x,y)f(y)\,\mathrm{d}\mathrm{v}_g(y), \qquad f\in L^2(\Sigma,\mathrm{d}\mathrm{v}_g),\] or equivalently by the conditions that, for each fixed \(x\in\Sigma\), \[\label{eq:Green-function-equations-boundary} \begin{cases} -\Delta_g G_{g,D}(x,\cdot)=2\pi\delta_x & \text{in }\Sigma,\\ G_{g,D}(x,\cdot)=0 & \text{on }\partial\Sigma. \end{cases}\tag{7}\] The function \(G_{g,\Sigma}\) is symmetric on \(\Sigma\times\Sigma\setminus\{(x,x):x\in\Sigma\}\).

In both cases, the Green function has the same local logarithmic singularity along the diagonal: \[G(x,y)=\log \frac{1}{d_g(x,y)}+O(1) \qquad\text{as }y\to x,\] where \(G\) stands either for \(G_g\) or for \(G_{g,D}\), depending on the situation.

2.4 Homology and cohomology↩︎

2.4.0.1 Closed surfaces

Consider a closed Riemann surface \((\Sigma,g)\) of genus \(\mathbb{g}\). Let \(H_1(\Sigma)\) denote the first homology group of \(\Sigma\) with value in \(\mathbb{Z}\). The algebraic intersection number endows \(H_1(\Sigma)\) with a symplectic structure. We will call a basis \(([a_i],[b_i])_{i=1,\ldots\mathbb{g}}\) of \(H_1(\Sigma)\) a geometric32symplectic32basis if the basis \(([a_i],[b_i])_{i=1,\ldots\mathbb{g}}\) is represented by simple closed curves \(a_1,b_1,\ldots,a_{\mathbb{g}},b_{\mathbb{g}}\) such that \(a_j\) intersects \(b_j\) transversely in one point, while all other intersections vanish.

We shall write \(\boldsymbol{\sigma}=(a_1,b_1,\dots,a_\mathbb{g},b_\mathbb{g})\) for such a basis, and also use the notation \(\boldsymbol{\sigma}=(\sigma_1,\dots,\sigma_{2\mathbb{g}})\) when no distinction between \(a\)- and \(b\)-cycles is needed.

Let \(H^1(\Sigma)\) denote the first de Rham cohomology space and \(\mathrm{Harm}^1(\Sigma)\) be the space of harmonic \(1\)-forms. The space \(H^1(\Sigma)\) is dual to the space \(H_1(\Sigma)\) via the pairing \[H^1(\Sigma)\times H_1(\Sigma)\rightarrow\mathbb{R}, \qquad (\omega,\sigma)\mapsto \langle\omega,\sigma\rangle=\int_\sigma\omega.\] We denote by \(H^1(\Sigma;\mathfrak{a})\) the first de Rham cohomology space associated to \(\mathfrak{a}\)-valued \(1\)-forms and define the space \(H^1_\Lambda(\Sigma;\mathfrak{a})\) of cohomology classes with periods in \(2\pi\Lambda\): \[H^1_\Lambda(\Sigma;\mathfrak{a}):= \left\{ \Omega\in H^1(\Sigma;\mathfrak{a}): \int_\sigma\Omega\in2\pi\Lambda \quad \forall\sigma\in H_1(\Sigma) \right\}.\]

2.4.0.2 Surfaces with boundary

We now turn to surfaces with boundary. Let \((\Sigma,[g])\) be a compact connected Riemann surface of genus \(\mathbb{g}\), with non-empty analytic boundary \(\partial\Sigma=\bigsqcup_{j=1}^{\mathbb{b}}\partial_j\Sigma\). Assume that the boundary components are oriented positively with respect to the orientation of \(\Sigma\), and we write \(c_j:=\partial_j\Sigma,\, j=1,\ldots,\mathbb{b}\) . As elements of \(H_1(\Sigma)\), they satisfy \([c_1]+\cdots+[c_{\mathbb{b}}]=[\partial\Sigma]=0.\)

The homology group \(H_1(\Sigma)\) is represented by oriented closed curves in \(\Sigma\) and is isomorphic to \(\mathbb{Z}^{2\mathbb{g}+\mathbb{b}-1}\). We call a basis \[(a_1,b_1,\ldots,a_{\mathbb{g}},b_{\mathbb{g}},c_1,\ldots,c_{\mathbb{b}-1})\] a canonical geometric basis of \(H_1(\Sigma)\) if \(a_i,b_i\) are simple closed curves contained in \(\Sigma^\circ\), chosen as in the closed case, and \(c_1,\ldots,c_{\mathbb{b}-1}\) are boundary cycles, after a possible renumbering of the boundary components.

The relative homology group \(H_1(\Sigma,\partial\Sigma)\) is represented by oriented closed curves in \(\Sigma^\circ\), together with oriented curves whose endpoints lie on \(\partial\Sigma\). It is also isomorphic to \(\mathbb{Z}^{2\mathbb{g}+\mathbb{b}-1}\). We call a basis \[(a_1,b_1,\ldots,a_{\mathbb{g}},b_{\mathbb{g}},d_1,\ldots,d_{\mathbb{b}-1})\] a canonical geometric basis of \(H_1(\Sigma,\partial\Sigma)\) if \(a_i,b_i\) are as above, and \(d_1,\ldots,d_{\mathbb{b}-1}\) are pairwise disjoint oriented simple arcs with endpoints on \(\partial\Sigma\), disjoint from the interior cycles \((a_i,b_i)\), such that the graph whose vertices are the boundary components and whose edges are the arcs \(d_j\) is connected and has no cycle.

We denote by \(H^1(\Sigma)\) and \(H^1(\Sigma,\partial\Sigma)\) respectively the absolute and relative de Rham cohomology spaces. Absolute cohomology is dual to absolute homology, and relative cohomology is dual to relative homology, through the pairings \[\label{eq:absolute-cohomology-duality} H^1(\Sigma)\times H_1(\Sigma)\to\mathbb{R}, \qquad (\omega,\sigma)\mapsto\int_\sigma\omega,\tag{8}\] and \[\label{eq:relative-cohomology-duality} H^1(\Sigma,\partial\Sigma)\times H_1(\Sigma,\partial\Sigma)\to\mathbb{R}, \qquad (\omega,\sigma)\mapsto\int_\sigma\omega.\tag{9}\] Equivalently, a relative cohomology class may be represented by a closed \(1\)-form whose pullback to \(\partial\Sigma\) vanishes. In what follows, we often choose relative representatives compactly supported in \(\Sigma^\circ\).

For \(\mathfrak{a}\)-valued forms, we define \[H^1_\Lambda(\Sigma;\mathfrak{a}) := \left\{ \Omega\in H^1(\Sigma;\mathfrak{a}): \int_\sigma\Omega\in2\pi\Lambda \quad \forall\sigma\in H_1(\Sigma) \right\},\] and \[H^1_\Lambda(\Sigma,\partial\Sigma;\mathfrak{a}) := \left\{ \Omega\in H^1(\Sigma,\partial\Sigma;\mathfrak{a}): \int_\sigma\Omega\in2\pi\Lambda \quad \forall\sigma\in H_1(\Sigma,\partial\Sigma) \right\}.\]

We shall use the following \(\mathfrak{a}\)-valued versions of the basic homological lemmas from [1]. whose proofs can be obtained componentwise: after fixing an orthonormal basis \((\varepsilon_1,\ldots,\varepsilon_r)\) of \(\mathfrak{a}\), one writes each \(\mathfrak{a}\)-valued form as a sum of scalar forms and applies the corresponding scalar statement in [1] to each component.

2.5 Equivariant \(\mathfrak{a}\)-valued functions and distributions↩︎

Let \(\Sigma\) be a surface with or without boundary, and let \(\beta_1=\dim H_1(\Sigma)\) denote its first Betti number. Fix a base point \(x_0\in \Sigma\), and let \(\pi:\widetilde{\Sigma}_{x_0}\to \Sigma\) be the universal cover of \(\Sigma\), with distinguished lift \(\widetilde{x}_0\) of \(x_0\). We view \(\widetilde{\Sigma}_{x_0}\) as the space of homotopy classes, with fixed endpoints, of continuous paths starting at \(x_0\). The fundamental group \(\pi_1(\Sigma,x_0)\) acts on \(\widetilde{\Sigma}_{x_0}\) by deck transformations.

Let \(\Omega\) be a smooth closed \(\mathfrak{a}\)-valued \(1\)-form on \(\Sigma\). For each \(\widetilde{x}\in \widetilde{\Sigma}_{x_0}\), define \[\label{eq:primitive-cover} I_{x_0}(\Omega)(\widetilde{x}) := \int_{\alpha_{x_0,x}}\Omega, \qquad x=\pi(\widetilde{x}),\tag{10}\] where \(\alpha_{x_0,x}\) is any \(C^1\) path from \(x_0\) to \(x\) whose lift joins \(\widetilde{x}_0\) to \(\widetilde{x}\). Since \(\Omega\) is closed, the quantity \(I_{x_0}(\Omega)(\widetilde{x})\) depends only on \(\widetilde{x}\), and therefore defines a smooth \(\mathfrak{a}\)-valued function on \(\widetilde{\Sigma}_{x_0}\). Moreover, \(\mathrm{d}I_{x_0}(\Omega)=\pi^*\Omega.\)

When \(\Omega\in H^1_\Lambda(\Sigma;\mathfrak{a})\), the primitive \(I_{x_0}(\Omega)\) is not in general invariant under deck transformations, but its variation is controlled by the period lattice \(2\pi\Lambda\).

The previous lemma shows that the primitive \(I_{x_0}(\Omega)\) is not invariant under deck transformations in general, but its increments are constrained to lie in \(2\pi\Lambda\). This naturally leads to considering \(\mathfrak{a}\)-valued functions on \(\widetilde{\Sigma}_{x_0}\) whose monodromy is prescribed by the lattice \(2\pi\Lambda\).

We let \(\Gamma:=\pi_1(\Sigma,x_0)\) and define \[C^\infty_\Gamma(\widetilde{\Sigma}_{x_0};\mathfrak{a}) := \left\{ u\in C^\infty(\widetilde{\Sigma}_{x_0};\mathfrak{a}):\forall h\in \Gamma, h^*u-u\in 2\pi\Lambda \right\}.\] We note that each \(u\in C^\infty_\Gamma(\widetilde{\Sigma}_{x_0};\mathfrak{a})\) induces a group morphism \[\chi_u:\Gamma\rightarrow2\pi\Lambda,\quad \chi_u(h):=h^*u-u.\] Moreover, each morphism corresponds to an element \(\Omega_\chi\) in \(H^1_\Lambda(\Sigma;\mathfrak{a})\), obtained from first fixing a basis \((\Omega_1,\ldots\Omega_{\beta_1})\) of \(H^1_\Lambda(\Sigma;\mathfrak{a})\), then choosing \(\boldsymbol{\lambda}=(\lambda_1,\ldots,\lambda_{\beta_1})\in\Lambda^{\beta_1}\) so that \(\chi(h)=\int_h\Omega_{\boldsymbol{\lambda}}\) for all \(h\in\Gamma\), and finally setting \(\Omega_\chi:=\Omega_{\boldsymbol{\lambda}}\). We write \(\chi_{\boldsymbol{\lambda}}\) for the morphism associated to \(\Omega_{\boldsymbol{\lambda}}.\)

For each group morphism \(\chi:\Gamma\to 2\pi\Lambda,\) we define \[C^\infty_\chi(\widetilde{\Sigma}_{x_0};\mathfrak{a}) := \left\{ u\in C^\infty_\Gamma(\widetilde{\Sigma}_{x_0};\mathfrak{a}):\forall h\in \Gamma,\; h^*u-u=\chi(h) \right\}.\] Then for \(u\in C^\infty_{\chi_{\boldsymbol{\lambda}}}(\widetilde{\Sigma}_{x_0};\mathfrak{a})\), \(u-I_{x_0}(\Omega_{\boldsymbol{\lambda}})\) is \(\Gamma\)-invariant and so descends to a smooth function on \(\Sigma.\) This implies that we can rewrite the space \(C^\infty_{\chi_{\boldsymbol{\lambda}}}(\widetilde{\Sigma}_{x_0};\mathfrak{a})\) as \[C^\infty_{\chi_{\boldsymbol{\lambda}}}(\widetilde{\Sigma}_{x_0};\mathfrak{a})=\{\pi^* f+I_{x_0}(\Omega_{\boldsymbol{\lambda}}):f\in C^\infty(\Sigma;\mathfrak{a})\}\] and also the space \(C^\infty_\Gamma(\widetilde{\Sigma}_{x_0};\mathfrak{a})\) as \[C^\infty_\Gamma(\widetilde{\Sigma}_{x_0};\mathfrak{a})=\cup_{\boldsymbol{\lambda}\in\Lambda^{\beta_1}}C^\infty_{\chi_{\boldsymbol{\lambda}}}(\widetilde{\Sigma}_{x_0};\mathfrak{a}).\] That is, each \(u\in C^\infty_\Gamma(\widetilde{\Sigma}_{x_0};\mathfrak{a})\) can be uniquely represented by \(\pi^*f+I_{x_0}(\Omega_{\boldsymbol{\lambda}})\) for some \(f\in C^\infty(\Sigma;\mathfrak{a})\) and \(\boldsymbol{\lambda}\in\Lambda^{\beta_1}.\)

Next we define the equivariant Sobolev spaces similarly. For \(s\in\mathbb{R}\), we define the \(\mathbb{Z}\)-module \[H^s_{\Gamma}(\widetilde{\Sigma}_{x_0};\mathfrak{a}):=\{f\in H^s_{\mathrm{loc}}(\widetilde{\Sigma}_{x_0};\mathfrak{a}):\forall h\in\Gamma, h^*f-f\in 2\pi\Lambda\},\] where \(H^s_{\mathrm{loc}}(\widetilde{\Sigma}_{x_0};\mathfrak{a})\) denotes the space of \(\mathfrak{a}\)-valued distributions which belong to \(H^s(U;\mathfrak{a})\) on every relatively compact open set \(U\subset \widetilde{\Sigma}_{x_0}\). Analogous to smooth \(\mathfrak{a}\)-valued functions, each \(u\in H^s_\Gamma(\widetilde{\Sigma}_{x_0};\mathfrak{a})\) can be uniquely represented by \(u=\pi^* f+I_{x_0}(\Omega_{\boldsymbol{\lambda}})\) for some \(f\in H^s(\Sigma;\mathfrak{a})\) and some \(\boldsymbol{\lambda}\in \Lambda^{\beta_1}\).

2.6 Magnetic backgrounds and boundary windings↩︎

Let \(\Sigma\) be a Riemann surface of genus \(\mathbb{g}\) with or without boundary. If \(\partial\Sigma\neq\emptyset\), we write \(\partial\Sigma=\bigsqcup_{\ell=1}^{\mathbb{b}}\partial_\ell\Sigma\) for its boundary components and assign to each boundary component a sign \(\varsigma_\ell\in\{\pm1\}\), with the convention that \(\varsigma_\ell=-1\) for an outgoing boundary component and \(\varsigma_\ell=+1\) for an incoming boundary component. Let \(\mathbf{z}=(z_1,\ldots,z_{n_{\mathfrak{m}}})\) be \(n_{\mathfrak{m}}\) pairwise distinct points in the interior of \(\Sigma\), to which we attach magnetic charges \(\mathbf{m}=(m_1,\ldots,m_{n_{\mathfrak{m}}})\in \Lambda^{n_{\mathfrak{m}}}\) such that \(\sum_{i=1}^{n_{\mathfrak{m}}}m_i=0.\) We denote by \(U_1,\ldots,U_{n_{\mathfrak{m}}}\) neighborhoods of \(z_1,\ldots,z_{n_{\mathfrak{m}}}\) and by \(\psi_j:\mathbb{D}\rightarrow U_j\) biholomorphisms such that \(\psi_j(0)=z_j\). In the unit disk \(\mathbb{D}\), we use the coordinate \(z=re^{i\theta}\) and denote by \(\mathrm{d}\theta\) the closed and coclosed 1-form in the pointed disk \(\mathbb{D}\setminus\{0\}\). If \(\partial\Sigma\neq\emptyset\), we also choose collar neighborhoods \(V_\ell\) of \(\partial_\ell\Sigma\) and biholomorphisms \(\psi_{n_{\mathfrak{m}}+\ell}:\{z\in\mathbb{C}:\delta<|z|\le 1\}\to V_\ell\) with \(\psi_{n_{\mathfrak{m}}+\ell}(\mathbb{T})=\partial_\ell\Sigma\) for some \(\delta<1\). Let \(\beta_1:=\dim H_1(\Sigma)\) be the first Betti number.

We now construct the magnetic background associated with the marked-point charges and, in the boundary case, with the boundary winding data.

We note that the form \(\nu_{\mathbf{z},\mathbf{m}}\in L^1(\Sigma)\) (resp. \(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}\in L^1(\Sigma)\)) does not belong to \(L^2(\Sigma)\), but we can still define an \(L^2\)-norm via regularization: for \(\omega=\sum_{\ell=1}^r\omega_\ell\varepsilon_\ell\in C^\infty(\Sigma\setminus\{\mathbf{z}\},T^*\Sigma\otimes\mathfrak{a})\) with \((\varepsilon_\ell)_{\ell=1,\ldots,r}\) an orthonormal basis of \(\mathfrak{a}\), let \[\left\lVert \omega \right\rVert^2_{g,\varepsilon}:=\sum^r_{\ell=1}\int_{\Sigma_{\mathbf{z},\varepsilon,g}}\omega_\ell\wedge*\omega_\ell,\] where \(\Sigma_{\mathbf{z},\varepsilon,g}:=\Sigma\setminus\bigcup_{j=1}^{n_{\mathfrak{m}}}B_g(z_j,\varepsilon)\) with \(B_g(z_j,\varepsilon)\) the geodesic ball centered at \(z_j\) with radius \(\varepsilon\) with respect to the metric \(g.\)

2.7 Equivariant \(\mathfrak{a}\)-valued functions and distributions with marked points↩︎

Let \((\Sigma,g)\) be a closed Riemann surface, let \(\mathbf{z}=(z_1,\ldots,z_{n_{\mathfrak{m}}})\) be pairwise distinct marked points, and set \(\Sigma_{\mathbf{z}}:=\Sigma\setminus\{z_1,\ldots,z_{n_{\mathfrak{m}}}\}.\) Fix a base point \(x_0\in\Sigma_{\mathbf{z}}\), and let \(\pi:\widetilde{\Sigma}_{\mathbf{z},x_0}\to \Sigma_{\mathbf{z}}\) be the universal cover, with distinguished lift \(\widetilde{x}_0\) of \(x_0\). We denote by \(\Gamma_{\mathbf{z}}:=\pi_1(\Sigma_{\mathbf{z}},x_0)\) the first fundamental group.

We first introduce the equivariant \(L^2\) space on the universal cover. We set \[L^2_{\Gamma_{\mathbf{z}}}(\widetilde{\Sigma}_{\mathbf{z},x_0};\mathfrak{a}) := \left\{ u\in L^2_{\mathrm{loc}}(\widetilde{\Sigma}_{\mathbf{z},x_0};\mathfrak{a}): \forall \gamma\in\Gamma_{\mathbf{z}},\; \gamma^*u-u\in 2\pi\Lambda \right\}.\] For each group morphism \(\chi:\Gamma_{\mathbf{z}}\to 2\pi\Lambda,\) we also write \[L^2_{\chi}(\widetilde{\Sigma}_{\mathbf{z},x_0};\mathfrak{a}) := \left\{ u\in L^2_{\mathrm{loc}}(\widetilde{\Sigma}_{\mathbf{z},x_0};\mathfrak{a}): \forall \gamma\in\Gamma_{\mathbf{z}},\; \gamma^*u-u=\chi(\gamma) \right\}.\] Thus \(L^2_{\Gamma_{\mathbf{z}}}(\widetilde{\Sigma}_{\mathbf{z},x_0};\mathfrak{a})=\cup_{\chi}L^2_{\chi}(\widetilde{\Sigma}_{\mathbf{z},x_0};\mathfrak{a}).\)

Now let \(\mathbf{m}=(m_1,\ldots,m_{n_{\mathfrak{m}}})\in\Lambda^{n_{\mathfrak{m}}},\) with \(\sum_{j=1}^{n_{\mathfrak{m}}}m_j=0,\) and let \(\nu_{\mathbf{z},\mathbf{m}}\) be the magnetic background \(1\)-form introduced in Proposition 4. Since \(\Omega_{\boldsymbol{\lambda}}+\nu_{\mathbf{z},\mathbf{m}}\), \(\boldsymbol{\lambda}\in\Lambda^{2\mathbb{g}}\) is closed on \(\Sigma_{\mathbf{z}}\), we can define its primitive on the universal cover by \[I_{x_0}(\Omega_{\boldsymbol{\lambda}}+\nu_{\mathbf{z},\mathbf{m}})(\widetilde{x}) := \int_{\alpha_{x_0,x}}(\Omega_{\boldsymbol{\lambda}}+\nu_{\mathbf{z},\mathbf{m}}), \qquad x=\pi(\widetilde{x}),\] where \(\alpha_{x_0,x}\) is any piecewise \(C^1\) path in \(\Sigma_{\mathbf{z}}\) from \(x_0\) to \(x\) whose lift joins \(\widetilde{x}_0\) to \(\widetilde{x}\).

By the period condition, this primitive is equivariant modulo \(2\pi\Lambda\), and it defines a group morphism \[\chi_{\boldsymbol{\lambda},\mathbf{m}}:\Gamma_{\mathbf{z}}\to 2\pi\Lambda, \qquad \chi_{\boldsymbol{\lambda},\mathbf{m}}(\gamma) := \int_{\gamma}(\Omega_{\boldsymbol{\lambda}}+\nu_{\mathbf{z},\mathbf{m}}).\] Moreover, if \(u\in L^2_{\chi_{\boldsymbol{\lambda},\mathbf{m}}}(\widetilde{\Sigma}_{\mathbf{z},x_0};\mathfrak{a}),\) then \(u-I_{x_0}(\Omega_{\boldsymbol{\lambda}}+\nu_{\mathbf{z},\mathbf{m}})\) is \(\Gamma_{\mathbf{z}}\)-invariant, hence it descends to a function \(f\) on \(\Sigma_{\mathbf{z}}\). Since the primitive \(I_{x_0}(\Omega_{\boldsymbol{\lambda}}+\nu_{\mathbf{z},\mathbf{m}})\) is smooth on the universal cover, this gives \(u=\pi^*f+I_{x_0}(\Omega_{\boldsymbol{\lambda}}+\nu_{\mathbf{z},\mathbf{m}}),\) for some \(f\in L^2(\Sigma;\mathfrak{a})\). Conversely, every function of this form belongs to \(L^2_{\chi_{\boldsymbol{\lambda},\mathbf{m}}}(\widetilde{\Sigma}_{\mathbf{z},x_0};\mathfrak{a}).\) Therefore \[L^2_{\chi_{\boldsymbol{\lambda},\mathbf{m}}}(\widetilde{\Sigma}_{\mathbf{z},x_0};\mathfrak{a}) = \left\{ \pi^*f+I_{x_0}(\Omega_{\boldsymbol{\lambda}}+\nu_{\mathbf{z},\mathbf{m}}):\; f\in L^2(\Sigma;\mathfrak{a}) \right\},\] and \[L^2_{\Gamma_{\mathbf{z}}}(\widetilde{\Sigma}_{\mathbf{z},x_0};\mathfrak{a}) = \bigcup_{\substack{\boldsymbol{\lambda}\in\Lambda^{2\mathbb{g}}\\ \mathbf{m}\in\Lambda^{n_{\mathfrak{m}}},\;\sum_{j=1}^{n_{\mathfrak{m}}}m_j=0}} \left\{ \pi^*f+I_{x_0}(\Omega_{\boldsymbol{\lambda}}+\nu_{\mathbf{z},\mathbf{m}}):\; f\in L^2(\Sigma;\mathfrak{a}) \right\}.\]

We now define the corresponding Sobolev space. For \(s\in(-\frac{1}{2},0)\), we set \[H^s_{\Gamma_{\mathbf{z}}}(\widetilde{\Sigma}_{\mathbf{z},x_0};\mathfrak{a}) := L^2_{\Gamma_{\mathbf{z}}}(\widetilde{\Sigma}_{\mathbf{z},x_0};\mathfrak{a})+\pi^*H^s(\Sigma;\mathfrak{a}).\] Equivalently, \[H^s_{\Gamma_{\mathbf{z}}}(\widetilde{\Sigma}_{\mathbf{z},x_0};\mathfrak{a}) = \bigcup_{\substack{\boldsymbol{\lambda}\in\Lambda^{2\mathbb{g}}\\ \mathbf{m}\in\Lambda^{n_{\mathfrak{m}}},\;\sum_{j=1}^{n_{\mathfrak{m}}}m_j=0}} \left\{ \pi^*f+I_{x_0}(\Omega_{\boldsymbol{\lambda}}+\nu_{\mathbf{z},\mathbf{m}}):\; f\in H^s(\Sigma;\mathfrak{a}) \right\}.\]

In other words, an element of \(H^s_{\Gamma_{\mathbf{z}}}(\widetilde{\Sigma}_{\mathbf{z},x_0};\mathfrak{a})\) is obtained by taking a Sobolev distribution \(f\in H^s(\Sigma;\mathfrak{a})\) on the closed surface, pulling it back to the universal cover, and then adding the fixed multivalued smooth background \(I_{x_0}(\Omega_{\boldsymbol{\lambda}}+\nu_{\mathbf{z},\mathbf{m}})\) carrying the topological and magnetic monodromy.

3 Imaginary Gaussian multiplicative chaos↩︎

In this section we recall the \(\mathfrak{a}\)-valued Gaussian free field on a compact Riemann surface \((\Sigma,g)\), with or without boundary, introduce its regularizations, and define the associated imaginary Gaussian multiplicative chaos. This is the probabilistic object that will later be used to make sense of the Toda interaction term.

We recall the notations from Section 2.3.

3.1 The \(\mathfrak{a}\)-valued Gaussian free field↩︎

Assume first that \(\partial\Sigma=\emptyset\). Let \((\varphi_j)_{j\in\mathbb{N}_0}\) be an orthonormal basis of real-valued eigenfunctions of \(\Delta_g\) in \(L^2(\Sigma,\mathrm{d}\mathrm{v}_g)\), with eigenvalues \[0=\lambda_0<\lambda_1\le \lambda_2\le \cdots,\] and with \(\varphi_0=\mathrm{v}_g(\Sigma)^{-1/2}.\) Let \((\varepsilon_1,\ldots,\varepsilon_r)\) be an orthonormal basis of \(\mathfrak{a}\), and let \((a_{j,\ell})_{j\ge1,\,1\le \ell\le r}\) be i.i.d.standard real Gaussian random variables. We define the \(\mathfrak{a}\)-valued Gaussian free field with vanishing mean in the metric \(g\) by \[\label{eq:aGFF-series} \mathbf{X}_g := \sqrt{2\pi}\sum_{j\ge1}\sum_{\ell=1}^r \frac{a_{j,\ell}}{\sqrt{\lambda_j}}\,\varphi_j\,\varepsilon_\ell.\tag{12}\] This series converges almost surely in \(H^s(\Sigma;\mathfrak{a})\) for every \(s<0\) (see [11], [12]), where \[H^s(\Sigma;\mathfrak{a}) = \left\{ u=\sum_{j\ge0}u_j\varphi_j:\; u_j\in \mathfrak{a},\quad \|u\|_{H^s(\Sigma;\mathfrak{a})}^2 := |u_0|^2+\sum_{j\ge1}\lambda_j^s|u_j|^2<\infty \right\}.\] Equivalently, for every \(u,v\in \mathfrak{a}\) and distinct points \(x,y\in\Sigma\), one has \[\mathbb{E}\big[\langle u,\mathbf{X}_g(x)\rangle\langle v,\mathbf{X}_g(y)\rangle\big] = \langle u,v\rangle\,G_g(x,y).\]

If now \(\partial\Sigma\neq\emptyset\), one defines analogously the \(\mathfrak{a}\)-valued Dirichlet Gaussian free field, denoted by \(\mathbf{X}_{g,D}\), by using the orthonormal basis of eigenfunctions of the Dirichlet Laplacian on \(\Sigma\). In that case, for every \(u,v\in \mathfrak{a}\) and distinct points \(x,y\in\Sigma\), \[\mathbb{E}\big[\langle u,\mathbf{X}_{g,D}(x)\rangle\langle v,\mathbf{X}_{g,D}(y)\rangle\big] = \langle u,v\rangle\,G_{g,D}(x,y),\] where \(G_{g,D}\) is the Dirichlet Green function introduced in Section 2.3.

We recall the Markov property of GFF, which is an essential ingredient in the proof of Segal’s gluing axioms (see Section 6).

3.2 \(g\)-regularization↩︎

Since the Gaussian free field is only a random distribution, we need to regularize it to define the exponential interaction terms. We do this by averaging over small geodesic circles.

Let \(h\) be a random distribution on \(\Sigma\). For \(x\in \Sigma\setminus\partial\Sigma\) and \(\varepsilon>0\) small enough so that the geodesic circle is well defined and contained in the interior of \(\Sigma\), let \(\mathcal{C}_g(x,\varepsilon)\) be the geodesic circle of center \(x\) and radius \(\varepsilon\), and let \(\mu_{x,\varepsilon}\) denote the uniform probability measure on \(\mathcal{C}_g(x,\varepsilon)\) induced by the metric \(g\). Choose a sequence \((f^n_{x,\varepsilon})_{n\ge1}\subset C^\infty(\Sigma)\) such that \[\|f^n_{x,\varepsilon}\|_{L^1(\Sigma,d\mathrm{v}_g)}=1, \qquad f^n_{x,\varepsilon}(y)=\theta_n\!\left(\frac{d_g(x,y)}{\varepsilon}\right),\] where each \(\theta_n\in C_c^\infty((0,2))\) is non-negative, supported near \(1\), and such that the measures \(f^n_{x,\varepsilon}\,d\mathrm{v}_g\) converge in \(\mathcal{D}'(\Sigma)\) to \(\mu_{x,\varepsilon}\) as \(n\to\infty\).

If the pairings \(\langle h,f^n_{x,\varepsilon}\rangle\) converge almost surely, as \(n\to\infty\), to a random variable \(h_\varepsilon(x)\), and if the resulting field \((x,\varepsilon)\mapsto h_\varepsilon(x)\) admits a continuous modification, then we say that \(h\) admits a \(g\)-regularization, and we call \(h_\varepsilon(x)\) the geodesic circle average regularization of \(h\). In particular, \(\mathbf{X}_g\) and \(\mathbf{X}_{g,D}\) admit such regularization.

For any compact set \(K\subset \Sigma\setminus\partial\Sigma\), one has [35] \[\mathbb{E}\big[\langle u,\mathbf{X}_{\varepsilon}(x)\rangle^2\big]=|u|^2(\log \varepsilon^{-1}+W(x))+o(1)\] uniformly in \(x\in K\), where \(\mathbf{X}=\mathbf{X}_g\) or \(\mathbf{X}_{g,D}\) and \[W(x):=\lim_{y\to x}\left(G(x,y)-\log\frac{1}{d_g(x,y)}\right),\]with the corresponding choice of the Green’s function, is called the Robin mass and is smooth on \(\Sigma\).

3.3 Imaginary Gaussian multiplicative chaos↩︎

Let \(\gamma\in\mathbb{R}\). If \(h\) is a random distribution admitting a \(g\)-regularization \((h_\varepsilon)_\varepsilon\), we define \[\label{eq:IGMC-eps} M^{g,\varepsilon}_{\gamma}(h,\mathrm{d}x) := \varepsilon^{-\gamma^2/2}e^{i\gamma h_\varepsilon(x)}\,\mathrm{d}\mathrm{v}_g(x).\tag{13}\]

In the Toda theory, the interaction is built from the simple-root projections of the field. Thus, for each simple root \(e_i\), \(i=1,\ldots,r\), we define for \(\mathbf{X}=\mathbf{X}_g\) or \(\mathbf{X}=\mathbf{X}_{g,D}\) \[\label{eq:IGMC-root-eps} M^{g,\varepsilon}_{\gamma e_i}(\mathbf{X},\mathrm{d}x) := \varepsilon^{-\frac{\gamma^2}{2}\langle e_i,e_i\rangle} e^{\boldsymbol{\mathrm{i}}\gamma\langle e_i,\mathbf{X}_{\varepsilon}(x)\rangle}\,\mathrm{d}\mathrm{v}_g(x).\tag{14}\] If \(\gamma^2\langle e_i,e_i\rangle<2,\) as \(\varepsilon\to0\), this measure converges in \(L^2(\Omega)\), weakly in the space of distributions, to a non-trivial random distribution of order \(2\) [36], denoted by \(M^g_{\gamma e_i}(\mathbf{X}_g,\mathrm{d}x)\). More precisely, for each \(i=1,\ldots,r\), there exists \(D_{\Sigma,i}\in L^2(\Omega)\) such that for every \(\varphi\in C^\infty(\Sigma)\) \[\label{eq:order-two-bound} \left| \int_\Sigma \varphi(x)\,M^g_{\gamma e_i}(\mathrm{d}x) \right| \le D_{\Sigma,i}\bigl(\|\varphi\|_\infty+\|\Delta_g\varphi\|_\infty\bigr) \qquad\text{a.s.}\tag{15}\] Moreover, if \(g'=e^\rho g\), then \[\label{eq:weyl32IGMC} M_{\gamma e_i}^{g'}(\mathbf{X},\mathrm{d}x) = e^{(1-\gamma^2\langle e_i,e_i\rangle/4)\rho(x)}\,M_{\gamma e_i}^g(\mathbf{X},\mathrm{d}x).\tag{16}\]

We note that under our normalization, the longest root has norm \(2\), and so the condition \(\gamma^2<1\) implies that \(\gamma^2\langle e_i,e_i\rangle<2, \,i=1,\ldots,r.\)

3.4 Exponential moments↩︎

The key integrability input for the Toda interaction is that imaginary GMC has exponential moments.

4 Path integrals and correlation functions↩︎

We now construct the path integral on a closed Riemann surface. We begin with the regularized curvature terms appearing in the action 1 , then define the path integral itself. We finally introduce correlation functions with electric and magnetic insertions and prove their basic covariance properties.

4.1 Curvature term↩︎

Let \((\Sigma,g)\) be a closed oriented Riemann surface. Since \(I_{x_0}(\Omega_{\boldsymbol{\lambda}})\) is a multivalued function on \(\Sigma\) and \(I_{x_0}(\nu_{\mathbf{z},\mathbf{m}})\) is a multivalued function on \(\Sigma\setminus\{\mathbf{z}\}\), we need to make sense of the curvature terms in the Toda action 1 , namely \[\int_\Sigma K_g I_{x_0}(\Omega_{\boldsymbol{\lambda}})\mathrm{d}\mathrm{v}_g, \qquad \int_{\Sigma\setminus\{\mathbf{z}\}} K_g I_{x_0}(\nu_{\mathbf{z},\mathbf{m}})\mathrm{d}\mathrm{v}_g.\] We regularize these terms by choosing branch cuts so that the corresponding primitives become single-valued on open dense subsets, and by adding correction terms supported on the cuts in order to recover the required invariance properties.

4.1.1 Curvature term associated to \(H^1_\Lambda(\Sigma;\mathfrak{a})\)↩︎

For \(\Omega\in H^1_\Lambda(\Sigma;\mathfrak{a})\), we define the period morphism \(\chi_\Omega: H_1(\Sigma)\rightarrow2\pi\Lambda\) by \[\chi_\Omega(\gamma):=\int_\gamma\Omega.\] If \(\boldsymbol{\sigma}=(a_j,b_j)_{j=1,\ldots,\mathbb{g}}\) is a geometric symplectic basis of \(H_1(\Sigma)\), we set \(\Sigma_{\boldsymbol{\sigma}}:=\Sigma\setminus\bigcup^{\mathbb{g}}_{j=1}(a_j\cup b_j).\) For \(x_0\in\Sigma_{\boldsymbol{\sigma}}\) a fixed base point and any closed \(\mathfrak{a}\)-valued \(1\)-form \(\Omega\), we define \[\label{primitive} I^{\boldsymbol{\sigma}}_{x_0}(\Omega)(x):=\int_{\alpha_{x_0,x}}\Omega,\qquad x\in\Sigma_{\boldsymbol{\sigma}},\tag{17}\] where \(\alpha_{x_0,x}\subset\Sigma_{\boldsymbol{\sigma}}\) is any smooth path from \(x_0\) to \(x\). This defines a smooth \(\mathfrak{a}\)-valued function on \(\Sigma_{\boldsymbol{\sigma}}\) with \(\mathrm{d}I^{\boldsymbol{\sigma}}_{x_0}(\Omega)=\Omega.\)

4.1.2 Curvature terms associated to magnetic points↩︎

Let \(\mathbf{z}=(z_1,\ldots,z_{n_{\mathfrak{m}}})\in\Sigma^{n_{\mathfrak{m}}}\) be pairwise distinct marked points and let \(\mathbf{m}=(m_1,\ldots,m_{n_{\mathfrak{m}}})\in\Lambda^{n_{\mathfrak{m}}}\) be magnetic charges satisfying \(\sum_{j=1}^{n_{\mathfrak{m}}} m_j=0\). Moreover, we let \(v_j\in T_{z_j}\Sigma,\, j=1,\ldots,n_{\mathfrak{m}},\) be unit tangent vectors and denote \[\mathbf{v}=((z_1,v_1),\ldots,(z_{n_{\mathfrak{m}}},v_{n_{\mathfrak{m}}}))\in (T\Sigma)^{n_{\mathfrak{m}}}.\] We shall use a system of branch cuts joining the marked points to make the magnetic primitive single-valued. We endow the lattice \(\Lambda\) with the lexicographic order induced by the ordered basis \((\omega^\vee_1,\ldots,\omega_r^\vee)\) and adapt [1] to our setting.

Let \(x_0\in \Sigma\setminus \mathcal{D}_{\mathbf{v},\boldsymbol{\xi}}\). By Lemma 10, we may define on \(\Sigma\setminus\mathcal{D}_{\mathbf{v},\boldsymbol{\xi}}\) \[I^{\boldsymbol{\xi}}_{x_0}(\nu_{\mathbf{z},\mathbf{m}})(x):=\int_{\alpha_{x_0,x}}\nu_{\mathbf{z},\mathbf{m}},\] where \(\alpha_{x_0,x}\) is any smooth curve in \(\Sigma\setminus\mathcal{D}_{\mathbf{v},\boldsymbol{\xi}}\) from \(x_0\) to \(x\).

4.2 Path integrals↩︎

Throughout the rest of this section, we work under the assumptions that \(\gamma^2<1\) and that \(Q\in\Lambda^*\) with \(\gamma>0\) so that both the Toda interaction characters and the curvature phase are compatible with the compactification lattice \(\Lambda\), where we recall that the background charge is given by \[Q=\gamma\rho-\frac{2}{\gamma}\rho^\vee.\]

We fix the following data:

• a geometric symplectic basis \(\boldsymbol{\sigma}=(\sigma_1,\ldots,\sigma_{2\mathbb{g}})\) of \(H_1(\Sigma)\) and closed smooth real-valued \(1\)-forms \(\eta_1,\ldots,\eta_{2\mathbb{g}}\) dual to \(\boldsymbol{\sigma}\) as in Lemma 1;

• a base point \(x_0\in \Sigma_{\boldsymbol{\sigma}}=\Sigma\setminus\cup^{2\mathbb{g}}_{j=1}\sigma_j\) and \(I_{x_0}^{\boldsymbol{\sigma}}(\Omega)\) the smooth function on \(\Sigma_{\boldsymbol{\sigma}}\) defined in 17

We first introduce the space of test functions on which the path integral is defined. Let \((e_j)_{j\geq 0}\) be an orthonormal basis of \(L^2(\Sigma,\mathrm{d}\mathrm{v}_g)\) consisting of eigenfunctions of \(\Delta_g\). We shall write elements of \(H^{s}(\Sigma;\mathfrak{a})\), \(s<0\) in the form \[f=f_0+\sqrt{2\pi}\sum_{j\geq 1} f_j e_j,\] and equip the Sobolev space \(H^{s}(\Sigma;\mathfrak{a})\) with the pushforward of the measure \(\mathrm{d}c\otimes \mathbb{P}\) on \(\mathbb{T}(\gamma)\times\Omega\) via the map \((c,\omega)\mapsto c+\mathbf{X}_g(\omega).\)

We recall that from Section 2.5, any equivariant distribution \(u\in H_\Gamma^{s}(\widetilde{\Sigma};\mathfrak{a})\) can be uniquely written as \[u=\pi^* (f_0+\sqrt{2\pi}\sum_{j\geq 1} f_j e_j)+I_{x_0}(\Omega_{\boldsymbol{\lambda}})\] for some \(f\in H^s(\Sigma;\mathfrak{a})\) and \(\boldsymbol{\lambda}\in \Lambda^{2\mathbb{g}}\simeq H_1(\Sigma)\). The space \(\mathcal{E}_\Lambda(\Sigma;\mathfrak{a})\) of test functions consists of all functionals \(F:H^s_\Gamma(\widetilde{\Sigma};\mathfrak{a})\rightarrow\mathbb{C}\) of the form \[\label{eq:test-functions-elementary} F(u) = \sum_{\ell\in A} e^{\boldsymbol{\mathrm{i}}\langle \ell,f_0\rangle} P_{\ell}(f-f_0)\, G_{\ell} \left( e^{\boldsymbol{\mathrm{i}}\langle q,I^{\boldsymbol{\sigma}}_{x_0}(\Omega_{\boldsymbol{\lambda}})\rangle} \right)\tag{18}\] for some finite subset \(A\subset\Lambda^*\) and \(q\in\Lambda^*\) where each \(P_{\ell}\), depending on \(\boldsymbol{\lambda},\) is a polynomial in the non-zero mode \(f-f_0\), namely \(P_{\ell}(f-f_0) = \mathcal{P}_{\ell} \left( \langle f-f_0,h_1\rangle,\ldots, \langle f-f_0,h_{m_\ell}\rangle \right),\) for some polynomial \(\mathcal{P}_{\ell,\boldsymbol{\lambda}}\) and \(h_1,\ldots,h_m\in H^{s}_0(\Sigma;\mathfrak{a})\), and each \(G_{\ell}\), also depending on \(\boldsymbol{\lambda}\), is a bounded continuous function on \(C^0(\Sigma_{\boldsymbol{\sigma}};\mathbb{S}^1).\) One can observe that the space is independent of the choice of the representative of \([\Omega_{\boldsymbol{\lambda}}]\in H^1(\Sigma;\mathfrak{a})\).

The condition \(\ell,q \in\Lambda^*\) ensures that \(F\) is \(2\pi\Lambda\)-periodic in the compactified field. In particular, \(F(u+2\pi\lambda)=F(u), \, \lambda\in\Lambda.\) The polynomial dependence on the non-zero mode is important: it is the class on which one can later perform analytic continuation arguments for correlation functions.

We now introduce the normed space obtained by completing these elementary test functions. For \(\boldsymbol{\lambda}\in\Lambda^{2\mathbb{g}}\), let \(\Omega_{\boldsymbol{\lambda}}^{\mathrm h} := \Pi_1\Omega_{\boldsymbol{\lambda}}\) be the harmonic representative of the cohomology class \([\Omega_{\boldsymbol{\lambda}}]\), and write \(\Omega_{\boldsymbol{\lambda}} = \Omega_{\boldsymbol{\lambda}}^{\mathrm h} + \mathrm{d}f_{\boldsymbol{\lambda}},\) where \(f_{\boldsymbol{\lambda}}\) is a smooth \(\mathfrak{a}\)-valued function. For \(p\geq 1\), define \[\label{eq:L-infty-p-norm} \|F\|_{\mathcal{L}^{\infty,p}_{\Lambda}} := \sup_{\boldsymbol{\lambda}\in\Lambda^{2\mathbb{g}}} \left( \int_{\mathfrak{a}/(2\pi\Lambda)} \mathbb{E}\left[ e^{-\frac{1}{2\pi}\langle \mathrm{d}\mathbf{X}_g,\Omega_{\boldsymbol{\lambda}}\rangle_2 -\frac{1}{4\pi}\|\mathrm{d}f_{\boldsymbol{\lambda}}\|_2^2} \left| F\left(c+\mathbf{X}_g+I^{\boldsymbol{\sigma}}_{x_0}(\Omega_{\boldsymbol{\lambda}})\right) \right|^p \right]\mathrm{d}c \right)^{1/p}.\tag{19}\] We denote by \(\mathcal{L}^{\infty,p}_{\Lambda}(\Sigma;\mathfrak{a})\) the completion of \(\mathcal{E}_{\Lambda}(\Sigma;\mathfrak{a})\) with respect to this norm, after identifying two test functions whose distance is zero.

On a closed surface \(\Sigma\), and for a topological sector \(\boldsymbol{\lambda}\in\Lambda^{2\mathbb{g}}\), we write the Toda field as \[\Phi^{\boldsymbol{\lambda}}_g := c+\mathbf{X}_g+I^{\boldsymbol{\sigma}}_{x_0}(\Omega_{\boldsymbol{\lambda}}).\] Here \(c\in\mathfrak{a}/(2\pi\Lambda)\), \(\mathbf{X}_g\) is the \(\mathfrak{a}\)-valued GFF with zero average, and \(I^{\boldsymbol{\sigma}}_{x_0}(\Omega_{\boldsymbol{\lambda}})\) is the primitive of the closed \(1\)-form \(\Omega_{\boldsymbol{\lambda}}\) on the cut surface \(\Sigma_{\boldsymbol{\sigma}}\). Thus \(\Phi^{\boldsymbol{\lambda}}_g\) is first defined as an element of \(H^s(\Sigma_{\boldsymbol{\sigma}};\mathfrak{a})\), for \(s<0\). However, because the jumps of \(I^{\boldsymbol{\sigma}}_{x_0}(\Omega_{\boldsymbol{\lambda}})\) across the cuts belong to \(2\pi\Lambda\), the lift of \(\Phi^{\boldsymbol{\lambda}}_g\) to the universal cover extends uniquely to an equivariant distribution on \(\widetilde{\Sigma}\). In other words, \(\Phi^{\boldsymbol{\lambda}}_g\in H^s_\Gamma(\widetilde{\Sigma};\mathfrak{a}),\) with monodromy determined by \(\boldsymbol{\lambda}\).

In what follows we shall not distinguish notationally between the representative of \(\Phi^{\boldsymbol{\lambda}}_g\) on the cut surface and its equivariant lift to \(\widetilde{\Sigma}\). Thus, whenever \(F\) is a test function on \(H^s_\Gamma(\widetilde{\Sigma};\mathfrak{a})\), the expression \(F(\Phi^{\boldsymbol{\lambda}}_g)\) means that \(F\) is evaluated on this equivariant extension. The field \(\Phi^{\boldsymbol{\lambda}}_g\) depends on the zero mode \(c\), on the GFF \(\mathbf{X}_g\), on the topological sector \(\boldsymbol{\lambda}\), and on the auxiliary choices \((\boldsymbol{\sigma},x_0)\). The dependence on these auxiliary choices will disappear from the path integral after summing over \(\boldsymbol{\lambda}\) and using the regularized curvature term.

4.3 Correlation functions↩︎

4.3.0.1 Magnetic operators

We now extend the preceding construction to correlation functions with magnetic operators. Let \(\mathbf{z}=(z_1,\ldots,z_n)\in \Sigma^n\) be a family of distinct marked points, to which we associate magnetic charges \(\mathbf{m}=(m_1,\ldots,m_n)\in\Lambda^n\) satisfying the neutrality condition \(\sum_{j=1}^n m_j=0.\) We also fix tangent vectors \(\mathbf{v}=((z_1,v_1),\ldots,(z_n,v_n))\in (T\Sigma)^n.\)

Let \(\boldsymbol{\xi}\) be a defect graph associated with the data \((\mathbf{v},\mathbf{m})\) and we recall from Proposition 4 the harmonic representative of the magnetic \(1\)-form \(\nu^{\mathrm{h}}_{\mathbf{z},\mathbf{m}}\) with monodromy \(2\pi m_j\) around \(z_j\). We denote by \(I^{\boldsymbol{\xi}}_{x_0}(\nu_{\mathbf{z},\mathbf{m}}^{\mathrm h})\) the corresponding primitive on the cut surface determined by the defect graph.

For \(\boldsymbol{\lambda}\in\Lambda^{2\mathbb{g}}\), we consider the Toda field \[\Phi^{\boldsymbol{\lambda},\mathbf{m}}_g := c+\mathbf{X}_g +I^{\boldsymbol{\sigma}}_{x_0}(\Omega_{\boldsymbol{\lambda}}) +I^{\boldsymbol{\xi}}_{x_0}(\nu_{\mathbf{z},\mathbf{m}}^{\mathrm h}).\] This field is defined on the surface cut along both the homology cuts \(\boldsymbol{\sigma}\) and the defect graph \(\boldsymbol{\xi}\), but can be equivalently viewed as an equivariant distribution on the universal cover of \(\Sigma_{\mathbf{z}}=\Sigma\setminus\{z_1,\ldots,z_n\}\), with monodromy determined both by \(\boldsymbol{\lambda}\) and by \(\mathbf{m}\).

We introduce the corresponding elementary test functions. The space \(\mathcal{E}_{\Lambda}^{\mathbf{m}}(\Sigma;\mathfrak{a})\) consists of all functionals \(F:H^s_{\Gamma,\mathbf{m}}(\widetilde{\Sigma}_{\mathbf{z}};\mathfrak{a})\rightarrow \mathbb{C}\) which, in the sector \(\boldsymbol{\lambda}\), can be written as \[\label{eq:magnetic-test-functions-elementary} F(u) = \sum_{\ell\in A} e^{\boldsymbol{\mathrm{i}}\langle \ell,f_0\rangle} P_{\ell}(f-f_0)\, G_{\ell} \left( e^{\boldsymbol{\mathrm{i}}\langle q, I^{\boldsymbol{\sigma}}_{x_0}(\Omega_{\boldsymbol{\lambda}}) + I^{\boldsymbol{\xi}}_{x_0}(\nu_{\mathbf{z},\mathbf{m}}^{\mathrm h}) \rangle} \right),\tag{21}\] where, as before, \(A\subset\Lambda^*\) is finite, \(q\in\Lambda^*\), each \(P_{\ell}\), depending on \((\boldsymbol{\lambda},\mathbf{m})\), is a polynomial in the non-zero mode \(f-f_0\), and each \(G_{\ell}\), depending on \((\boldsymbol{\lambda},\mathbf{m})\) is a bounded continuous function on \(C^0(\Sigma_{\boldsymbol{\sigma},\boldsymbol{\xi}};\mathbb{S}^1).\)

The conditions \(\ell,q\in\Lambda^*\) ensure that the functional is well-defined on the compactified field and is invariant under addition of \(2\pi\Lambda\). Moreover, the space \(\mathcal{E}_\Lambda^{\mathbf{m}}(\Sigma;\mathfrak{a})\) does not depend on the choice of the representative \(\Omega_{\boldsymbol{\lambda}}+\nu^{\mathrm{h}}_{\mathbf{z},\mathbf{m}}\) of its cohomology class.

As before, we define the corresponding \(\mathcal{L}^{\infty,p}\)-norm for \(p\geq 1\) by \[\label{eq:L-infty-p-magnetic-norm} \begin{align} \|F\|_{\mathcal{L}^{\infty,p}_{\Lambda,\mathbf{m}}}:=\sup_{\boldsymbol{\lambda}\in\Lambda^{2\mathbb{g}}} \Bigg(\int_{\mathfrak{a}/(2\pi\Lambda)}\mathbb{E}\Big[ &e^{-\frac{1}{2\pi}\langle \mathrm{d}\mathbf{X}_g,\Omega_{\boldsymbol{\lambda}}+\nu_{\mathbf{z},\mathbf{m}}^{\mathrm h}\rangle_2 -\frac{1}{4\pi}\|\mathrm{d}f_{\boldsymbol{\lambda}}\|_2^2} \\&\times \left|F\left(c+\mathbf{X}_g +I^{\boldsymbol{\sigma}}_{x_0}(\Omega_{\boldsymbol{\lambda}}) +I^{\boldsymbol{\xi}}_{x_0}(\nu_{\mathbf{z},\mathbf{m}}^{\mathrm h}) \right)\right|^p\Big]\mathrm{d}c\Bigg)^{1/p}. \end{align}\tag{22}\] Here, as in the previous subsection, \(\Omega_{\boldsymbol{\lambda}} = \Omega_{\boldsymbol{\lambda}}^{\mathrm h} + \mathrm{d}f_{\boldsymbol{\lambda}}\) and we denote by \(\mathcal{L}^{\infty,p}_{\Lambda,\mathbf{m}}(\Sigma;\mathfrak{a})\) the completion of \(\mathcal{E}_{\Lambda,\mathbf{m}}(\Sigma;\mathfrak{a})\) with respect to this norm.

Following the same lines as in the proof of Lemma 13, we have:

4.3.0.2 Electric operators

We now construct electric operators in the presence of the magnetic background introduced above. The purely electric case is obtained by taking \(\mathbf{m}=0\).

Let \(x\in\Sigma_{\mathbf{z}}:=\Sigma\setminus\{z_1,\ldots,z_{n_{\mathfrak{m}}}\}\) and let \(\alpha\in\mathfrak{a}\). For an equivariant field \(u\in H^s_{\Gamma,\mathbf{m}}(\widetilde{\Sigma}_{\mathbf{z}};\mathfrak{a})\), we define the regularized electric operator by \[V^g_{\alpha,\epsilon}(u,x) := \epsilon^{-\frac{|\alpha|^2}{2}} e^{\boldsymbol{\mathrm{i}}\left\langle \alpha,u_{g,\epsilon}(x)\right\rangle},\] where \(u_{g,\epsilon}\) denotes a \(g\)-regularization of \(u\) at scale \(\epsilon\). When \(u=\Phi_g^{\boldsymbol{\lambda},\mathbf{m}}\), we simply write \[V^g_{\alpha,\epsilon}(x) := V^g_{\alpha,\epsilon}(\Phi_g^{\boldsymbol{\lambda},\mathbf{m}},x).\]

Let \(\mathbf{x}=(x_1,\ldots,x_{n_{\mathfrak{e}}})\in \Sigma_{\mathbf{z}}^{n_{\mathfrak{e}}}\) be distinct points, also distinct from the magnetic insertion points \(\mathbf{z}\), and let \(\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_{n_{\mathfrak{e}}})\in(\Lambda^*)^{n_{\mathfrak{e}}}\) be electric charges. We set \[V^{g,\epsilon}_{(\boldsymbol{\alpha},0)}(u,\mathbf{x}) := \prod_{j=1}^{n_{\mathfrak{e}}} V^g_{\alpha_j,\epsilon}(u,x_j),\] and again write \(V^{g,\epsilon}_{(\boldsymbol{\alpha},0)}(\mathbf{x})\) when the field is \(\Phi_g^{\boldsymbol{\lambda},\mathbf{m}}\). The condition \(\alpha_j\in\Lambda^*\) ensures that the operator is well-defined on the compactified field, namely it is invariant under the addition of \(2\pi\Lambda\) to the zero mode.

We introduce the electric shift \[u_{\mathbf{x}}(y):=\boldsymbol{\mathrm{i}}\sum_{j=1}^{n_{\mathfrak{e}}}\alpha_j G_g(y,x_j),\qquad y\in\Sigma,\] where \(G_g\) is the Green function of \(\Delta_g\) with zero average. Notice that \(u_{\mathbf{x}}\in H^s(\Sigma;\mathfrak{a}_{\mathbb{C}})\) for every \(s<1\). This will be the shift that appears after applying the Girsanov transform to the regularized electric insertions.

We now define the corresponding electric-magnetic norm. For \(p\geq1\), set \[\label{eq:L-infty-p-electric-magnetic-norm} \begin{align} \|F\|_{\mathcal{L}^{\infty,p}_{\Lambda,\boldsymbol{\alpha},\mathbf{m}}} := \sup_{\boldsymbol{\lambda}\in\Lambda^{2\mathbb{g}}} \Bigg( \int_{\mathfrak{a}/(2\pi\Lambda)} \mathbb{E}\Big[ & e^{-\frac{1}{2\pi} \langle \mathrm{d}\mathbf{X}_g,\Omega_{\boldsymbol{\lambda}}+\nu_{\mathbf{z},\mathbf{m}}^{\mathrm h}\rangle_2 -\frac{1}{4\pi}\|\mathrm{d}f_{\boldsymbol{\lambda}}\|_2^2} \\ &\times \left| F\left( c+\mathbf{X}_g +u_{\mathbf{x}} +I^{\boldsymbol{\sigma}}_{x_0}(\Omega_{\boldsymbol{\lambda}}) +I^{\boldsymbol{\xi}}_{x_0}(\nu_{\mathbf{z},\mathbf{m}}^{\mathrm h}) \right) \right|^p \Big] \mathrm{d}c \Bigg)^{1/p}. \end{align}\tag{25}\] Here \(\Omega_{\boldsymbol{\lambda}} = \Omega_{\boldsymbol{\lambda}}^{\mathrm h} + \mathrm{d}f_{\boldsymbol{\lambda}}\) as before. We denote by \(\mathcal{L}^{\infty,p}_{\Lambda,\boldsymbol{\alpha},\mathbf{m}}(\Sigma;\mathfrak{a})\) the completion of \(\mathcal{E}^\mathbf{m}_\Lambda(\Sigma;\mathfrak{a})\) with respect to this norm.

As in Lemma 14, the norm is independent of the choice of representative of the cohomology class of \(\Omega_{\boldsymbol{\lambda}}\):

For \(F\in\mathcal{E}^\mathbf{m}_\Lambda(\Sigma;\mathfrak{a})\), we define the regularized electric-magnetic correlation by \[\left\langle F V^{g,\epsilon}_{(\boldsymbol{\alpha},0)}(\mathbf{x}) V^g_{(0,\mathbf{m})}(\mathbf{v}) \right\rangle_{\Sigma,g} := \left\langle F\,V^{g,\epsilon}_{(\boldsymbol{\alpha},0)}(\Phi_g^{\boldsymbol{\lambda},\mathbf{m}},\mathbf{x}) V^g_{(0,\mathbf{m})}(\mathbf{v}) \right\rangle_{\Sigma,g},\] where the right-hand side is understood through Definition 5, with the regularized electric factor inserted in the expectation.

The path integral with electric and magnetic insertions is then defined, if the limit exists, by \[\label{eq:electric-magnetic-path-integral-limit} \left\langle F V^g_{(\boldsymbol{\alpha},0)}(\mathbf{x}) V^g_{(0,\mathbf{m})}(\mathbf{v}) \right\rangle_{\Sigma,g} := \lim_{\epsilon\to0} \left\langle F V^{g,\epsilon}_{(\boldsymbol{\alpha},0)}(\mathbf{x}) V^g_{(0,\mathbf{m})}(\mathbf{v}) \right\rangle_{\Sigma,g}.\tag{26}\]

4.3.0.7 Electro-magnetic operators

We set \(n_{\mathfrak{e}}=n_{\mathfrak{m}}\) and define the path integral with electro-magnetic operators by \[\langle FV^g_{(\boldsymbol{\alpha},\mathbf{m})}(\mathbf{v})\rangle_{\Sigma,g}:=\lim_{t\rightarrow1}\langle FV^g_{(\boldsymbol{\alpha},0)}(\mathbf{x}(t))V^g_{(0,\mathbf{m})}(\mathbf{v})\rangle_{\Sigma,g}\] for \(F\in\mathcal{E}_{\Lambda}^{\mathbf{m}}(\Sigma;\mathfrak{a})\), where \(\mathbf{x}(t)=(x_1(t),\ldots,x_{n_{\mathfrak{m}}}(t))\) with \(x_j:[0,1]\rightarrow \Sigma\) any \(C^1\) curve such that \(x_j(1)=z_j\) and \(\dot{x}_j(t)=v_j.\)

5 Correlation functions on the Riemann sphere↩︎

In this section, we apply Theorem 10 to write \(n\)-point correlation functions on the Riemann sphere \(\hat{\mathbb{C}}\) as a Dotsenko–Fateev type integral. We consider the metric \[g_0=|z|_+^{-4}|\mathrm{d}z|^2, \qquad |z|_+=\max\{|z|,1\}.\] Let \(\mathbf{z}=(z_1,\ldots,z_n)\in\hat{\mathbb{C}}\) be \(n\) pairwise distinct points and \(v_1,\ldots,v_n\) be the associated unit tangent vectors. Let \(\mathbf{m}=(m_1,\ldots,m_n)\in \Lambda\) be magnetic charges associated to \(\mathbf{z}\) satisfying \(\sum_{j=1}^nm_j=0.\)

As in the rank one case, the zero mode integral shows that the correlation functions vanish unless \[\label{eq:sphere-neutrality} \sum_{j=1}^n\alpha_j-2Q\in -\gamma\sum_{i=1}^r \mathbb{N}e_i.\tag{39}\] In particular, if \(\alpha_j-Q\in\mathcal{C}_+\) for all \(j\), then non-trivial correlation functions on the sphere can only start at \(n\ge 3\). Whenever 39 holds, we write \[\label{eq:sphere-screening-vector-n} 2Q-\sum_{j=1}^n\alpha_j=\gamma\sum_{i=1}^r s_i e_i, \qquad \mathbf{s}=(s_1,\ldots,s_r)\in\mathbb{N}^r.\tag{40}\]

Fix a defect graph \(\boldsymbol{\xi}\) associated with \((\mathbf{v},\mathbf{m})\), and let \(I^{\boldsymbol{\xi}}_0(\nu^{\mathrm h}_{\mathbf{z},\mathbf{m}})\) be the corresponding magnetic primitive, normalized to vanish near infinity. Using the formula \[G_{g_0}(x,y)=\log\frac{1}{|x-y|}+\log|x|_+ +\log|y|_+,\] performing the series expansion of the exponential in the expectation, and computing the Fourier coefficients, we obtain \[\label{eq:sphere-n-point-screened} \begin{align} &\langle V^{g_0}_{(\alpha_1,m_1)}(z_1)\cdots V^{g_0}_{(\alpha_n,m_n)}(z_n)\rangle_{\hat{\mathbb{C}},g_0} \\ &= \mathrm{Vol}(\mathbb{T}(\gamma)) \left(\frac{\mathrm{v}_{g_0}(\hat{\mathbb{C}})}{\det'(\Delta_{g_0})}\right)^{r/2} \prod_{i=1}^r \frac{(-\mu_i)^{s_i}}{s_i!} \prod_{1\le j<k\le n}|z_j-z_k|^{\langle\alpha_j,\alpha_k\rangle} \prod_{j=1}^n |z_j|_+^{4\Delta_{(\alpha_j,0)}} \\ &\quad\times e^{-\frac{1}{4\pi}\|\nu^{\mathrm h}_{\mathbf{z},\mathbf{m}}\|_{g_0,0}^2 +\boldsymbol{\mathrm{i}}\sum_{j=1}^n\langle\alpha_j,I^{\boldsymbol{\xi}}_0(\nu^{\mathrm h}_{\mathbf{z},\mathbf{m}})(z_j)\rangle} \mathbb{E}\Bigg[ \prod_{i=1}^r \left( \int_{\mathbb{C}} F_i(x,\mathbf{z})\,M^{g_0}_{\gamma e_i}(\mathbf{X}_{g_0},\mathrm{d}x) \right)^{s_i} \Bigg]. \end{align}\tag{41}\] where \[\label{eq:Fi-general-n} F_i(x,\mathbf{z}) := \prod_{j=1}^n \left( \frac{|x-z_j|}{|x|_+} \right)^{\gamma\langle e_i,\alpha_j\rangle} e^{\boldsymbol{\mathrm{i}}\gamma\langle e_i,I^{\boldsymbol{\xi}}_0(\nu^{\mathrm h}_{\mathbf{z},\mathbf{m}})(x)\rangle}.\tag{42}\]

We now rewrite the expectation term in 41 as a multiple integral.

Applying Theorem 10, we obtain that the correlation functions are conformally covariant: if \(g=e^\rho g_0=g(z)|\mathrm{d}z|^2\) is a conformal metric, if \(z_1,\ldots,z_n\) are distinct points on \(\hat{\mathbb{C}}\) with associated unit tangent vectors \(v_1,\ldots,v_n\), and if \(\psi(z)=\frac{az+b}{cz+d}\) is a Möbius map with \(ad-bc=1\), then \[\label{eq:sphere-mobius-cov} \langle V^g_{(\boldsymbol{\alpha},\mathbf{m})}(\psi_*\mathbf{v})\rangle_{\hat{\mathbb{C}},g} = \prod_{j=1}^n \left( \frac{|\psi'(z_j)|^2g(\psi(z_j))}{g(z_j)} \right)^{-\Delta_{(\alpha_j,m_j)}} \langle V^g_{(\boldsymbol{\alpha},\mathbf{m})}(\mathbf{v})\rangle_{\hat{\mathbb{C}},g}.\tag{47}\]

In particular, if \(n=3\) and \(m_1\leq m_2\leq m_3,\) then the Möbius covariance implies that the three-point function is determined up to a constant, denoted by \(C_{\gamma,\boldsymbol{\mu}}(\boldsymbol{\alpha},\mathbf{m})\) and called the structure constant. More precisely, \[\label{eq:sphere-three-point-structure} \begin{align} \langle V^g_{(\boldsymbol{\alpha},\mathbf{m})}(\mathbf{v})\rangle_{\hat{\mathbb{C}},g} =&e^{\frac{c}{96\pi}\int_{\hat{\mathbb{C}}} (|\mathrm{d}\rho|_{g_0}^2+2K_{g_0}\rho)\,\mathrm{d}\mathrm{v}_{g_0}} e^{-\boldsymbol{\mathrm{i}}\sum_{j=1}^3\langle Q,m_j\rangle\arg(v_j)} \\&\times P_{(\boldsymbol{\alpha},\mathbf{m})}(\mathbf{z}) \prod_{j=1}^3 g(z_j)^{-\Delta_{(\alpha_j,m_j)}} \,C_{\gamma,\boldsymbol{\mu}}(\boldsymbol{\alpha},\mathbf{m}), \end{align}\tag{48}\] where \[\begin{align} P_{(\boldsymbol{\alpha},\mathbf{m})}(\mathbf{z}) :=& |z_1-z_3|^{2(\Delta_{(\alpha_2,m_2)}-\Delta_{(\alpha_1,m_1)}-\Delta_{(\alpha_3,m_3)})} |z_2-z_3|^{2(\Delta_{(\alpha_1,m_1)}-\Delta_{(\alpha_2,m_2)}-\Delta_{(\alpha_3,m_3)})}\\&\times |z_1-z_2|^{2(\Delta_{(\alpha_3,m_3)}-\Delta_{(\alpha_1,m_1)}-\Delta_{(\alpha_2,m_2)})}. \end{align}\] In particular, if \(e_1=\partial_x\) in the coordinate \(z=x+\boldsymbol{\mathrm{i}}y\in\mathbb{C}\), then \[\begin{align} C_{\gamma,\boldsymbol{\mu}}(\boldsymbol{\alpha},\mathbf{m})=&\langle V_{(\alpha_1,m_1)}(0,e_1)V_{(\alpha_2,m_2)}(1,e_1)V_{(\alpha_3,m_3)}(\infty,e_1) \rangle_{\hat{\mathbb{C}},g_0}.\end{align}\]

We now compute the three-point function in the metric \(g_0\). Assume that \(z_1,z_2,z_3\in \mathbb{R}\) with \(z_1<z_2<z_3\) and that \(v_1=v_2=v_3=e_1\). We choose the defect graph \(z_1\to z_2\to z_3\), with branch cut \([z_1,z_3]\). We then take

\[\begin{align}\nu^{\mathrm{h}}_{\mathbf{z},\mathbf{m}}(z)=&\mathrm{Im}\left(\frac{m_1(z_1-z_2)\mathrm{d}z}{(z-z_1)(z-z_2)}+\frac{m_3(z_3-z_2)\mathrm{d}z}{(z-z_2)(z-z_3)}\right)\\ =&\mathrm{Im}\left((\frac{m_1}{z-z_1}+\frac{m_2}{z-z_2}+\frac{m_3}{z-z_3})\mathrm{d}z\right)\end{align}\] so that \(I^{\boldsymbol{\xi}}_0(\nu^{\mathrm h}_{\mathbf{z},\mathbf{m}})(x) = -m_1\,\arg\frac{z_2-x}{z_1-x} + m_3\,\arg\frac{z_3-x}{z_2-x},\) which vanishes on \(]z_3,+\infty[\).

Set \(\mathbf{s}=(s_1,\ldots,s_r)\in\mathbb{N}^r\) by \[2Q-\sum_{j=1}^3\alpha_j=\gamma\sum_{i=1}^r s_i e_i.\] Combining 41 , 43 and the observation [1] that \[e^{-\frac{1}{4\pi}\|\nu^{\mathrm h}_{\mathbf{z},\mathbf{m}}\|_{g_0,0}^2 +\boldsymbol{\mathrm{i}}\sum_{j=1}^3\langle\alpha_j,I^{\boldsymbol{\xi}}_0(\nu^{\mathrm h}_{\mathbf{z},\mathbf{m}})(z_j)\rangle} = \prod_{1\le j<k\le 3}|z_j-z_k|^{\langle m_j,m_k\rangle}\prod_{j=1}^3|z_j|_+^{|m_j|^2} e^{\boldsymbol{\mathrm{i}}\pi\langle\alpha_2,m_1\rangle-\boldsymbol{\mathrm{i}}\pi\langle\alpha_3,m_3\rangle},\] we obtain

\[\label{eq:sphere-three-point-DF} \begin{align} \langle V^{g_0}_{\boldsymbol{\alpha},\mathbf{m}}(\mathbf{z})\rangle_{\hat{\mathbb{C}},g_0} =&\mathrm{Vol}(\mathbb{T}(\gamma)) \left(\frac{\mathrm{v}_{g_0}(\hat{\mathbb{C}})}{\det'(\Delta_{g_0})}\right)^{r/2} \prod_{i=1}^r \frac{(-\mu_i)^{s_i}}{s_i!} \prod_{1\le j<k\le 3}|z_j-z_k|^{\langle\alpha_j,\alpha_k\rangle+\langle m_j,m_k\rangle} \\ &\times \prod_{j=1}^3|z_j|_+^{4\Delta_{(\alpha_j,m_j)}} e^{\boldsymbol{\mathrm{i}}\pi\langle\alpha_2,m_1\rangle-\boldsymbol{\mathrm{i}}\pi\langle\alpha_3,m_3\rangle} \mathcal{I}_{\mathbf{s}}(\boldsymbol{\alpha},\mathbf{m};\mathbf{z}). \end{align}\tag{49}\] where \[\label{eq:sphere-three-point-DF-integral} \begin{align} \mathcal{I}_{\mathbf{s}}(\boldsymbol{\alpha},\mathbf{m};\mathbf{z}) :={}& \int_{\mathbb{C}^N} \prod_{i=1}^r\prod_{a=1}^{s_i} (x_a^{(i)}-z_1)^{A_{i,1}} (\overline{x_a^{(i)}-z_1})^{\bar A_{i,1}} (x_a^{(i)}-z_2)^{A_{i,2}} (\overline{x_a^{(i)}-z_2})^{\bar A_{i,2}} \\ &\times (x_a^{(i)}-z_3)^{A_{i,3}} (\overline{x_a^{(i)}-z_3})^{\bar A_{i,3}} \prod_{i=1}^r\prod_{1\le a<b\le s_i} |x_a^{(i)}-x_b^{(i)}|^{\gamma^2\langle e_i,e_i\rangle} \\ &\times \prod_{1\le i<j\le r}\prod_{a=1}^{s_i}\prod_{b=1}^{s_j} |x_a^{(i)}-x_b^{(j)}|^{\gamma^2\langle e_i,e_j\rangle} \prod_{i=1}^r\prod_{a=1}^{s_i}\mathrm{d}x_a^{(i)}, \end{align}\tag{50}\] with \[A_{i,j}:=\frac{\gamma}{2}\langle e_i,\alpha_j+m_j\rangle, \qquad \bar A_{i,j}:=\frac{\gamma}{2}\langle e_i,\alpha_j-m_j\rangle, \qquad j=1,2,3.\]

The structure constant is then given by \[\label{eq:three-point-0-1-infty} \begin{align} C_{\gamma, ,\boldsymbol{\mu}}(\boldsymbol{\alpha},\mathbf{m}) =& \lim_{|z|\to\infty} \langle V^{g_0}_{(\alpha_1,m_1)}(0)V^{g_0}_{(\alpha_2,m_2)}(1)V^{g_0}_{(\alpha_3,m_3)}(z)\rangle_{\hat{\mathbb{C}},g_0}\\ =& \mathrm{Vol}(\mathbb{T}(\gamma)) \left(\frac{\mathrm{v}_{g_0}(\hat{\mathbb{C}})}{\det'(\Delta_{g_0})}\right)^{r/2} \prod_{i=1}^r \frac{(-\mu_i)^{s_i}}{s_i!}\, e^{i\pi\langle\alpha_2,m_1\rangle-i\pi\langle\alpha_3,m_3\rangle}\, \mathcal{I}_{\mathbf{s}}(\boldsymbol{\alpha},\mathbf{m}), \end{align}\tag{51}\] where \[\label{eq:DF-0-1-infty} \begin{align} \mathcal{I}_{\mathbf{s}}(\boldsymbol{\alpha},\mathbf{m}) :={}& \int_{\mathbb{C}^N} \prod_{i=1}^r\prod_{a=1}^{s_i} \bigl(x_a^{(i)}\bigr)^{\frac{\gamma}{2}\langle e_i,\alpha_1+m_1\rangle} \bigl(\bar x_a^{(i)}\bigr)^{\frac{\gamma}{2}\langle e_i,\alpha_1-m_1\rangle} \\ &\qquad\qquad\qquad\times \bigl(1-x_a^{(i)}\bigr)^{\frac{\gamma}{2}\langle e_i,\alpha_2+m_2\rangle} \bigl(1-\bar x_a^{(i)}\bigr)^{\frac{\gamma}{2}\langle e_i,\alpha_2-m_2\rangle} \\ &\qquad\qquad\qquad\times \prod_{i=1}^r\prod_{1\le a<b\le s_i} |x_a^{(i)}-x_b^{(i)}|^{\gamma^2\langle e_i,e_i\rangle} \\ &\qquad\qquad\qquad\times \prod_{1\le i<j\le r}\prod_{a=1}^{s_i}\prod_{b=1}^{s_j} |x_a^{(i)}-x_b^{(j)}|^{\gamma^2\langle e_i,e_j\rangle} \prod_{i=1}^r\prod_{a=1}^{s_i}\mathrm{d}x_a^{(i)}. \end{align}\tag{52}\]

In [20], when \(\mathfrak{g}=\mathfrak{sl}_{r+1}\), the integral 52 was computed in the pure electric and semidegenerate case, i.e., \(\mathbf{m}=0\) and \(\alpha_1=\kappa\omega_\gamma\). Following the argument in [20], we extend this result to the case where \(\alpha_1=\kappa\omega_r\) and \(m_1=0\).

To present the formula, we write \(l(x)=\Gamma(x)/\Gamma(1-x)\) with \(\Gamma\) the usual Gamma function and introduce the special function \(\Upsilon,\) defined intially on \(0<\mathrm{Re}(z)<q:=\gamma+\frac{2}{\gamma},\) where it admits the integral representation \[\label{eq:Upsilon} \log\Upsilon(z)=\int^\infty_0\left(\left(\frac{q}{2}-z\right)^2\frac{e^{-t}}{2}-\frac{\left(\sinh\left(\left(\frac{q}{2}-z\right)\frac{t}{2\sqrt{2}}\right)\right)^2}{\sinh\left(\frac{t\gamma}{2\sqrt{2}}\right)\sinh\left(\frac{t}{\sqrt{2}\gamma}\right)}\right)\frac{\mathrm{d}t}{t},\tag{53}\] and then extended analytically to an entire function via the shift equations \[\label{eq:shift} \Upsilon(z+\chi)=l\left(\frac{\chi}{2}z\right)\left(\frac{\chi}{\sqrt{2}}\right)^{1-\chi z}\Upsilon(z),\quad \chi\in\left\{\gamma,\frac{2}{\gamma}\right\}.\tag{54}\] We note that if \(\gamma^2\notin\mathbb{Q}\), the special function \(\Upsilon\) has no poles but only simple zeros given by \((-\gamma\mathbb{N}-\frac{2}{\gamma}\mathbb{N})\cup(q+\gamma\mathbb{N}+\frac{2}{\gamma}\mathbb{N})\).

In the case where \(\alpha_1=\kappa\omega_r\) and \(\mathbf{m}=0\), the structure constant can be written as \[C_{\gamma,\boldsymbol{\mu}}(\kappa\omega_r,\alpha_2,\alpha_3,0,0,0)=\mathrm{Vol}(\mathbb{T}(\gamma)) \left(\frac{\mathrm{v}_{g_0}(\hat{\mathbb{C}})}{\det'(\Delta_{g_0})}\right)^{r/2} \prod_{i=1}^r (-\pi\mu_i)^{s_i}C^{\mathrm{FL}}_\gamma(\kappa\omega_r,\alpha_2,\alpha_3),\] where \(C^{\mathrm{FL}}_\gamma(\kappa\omega_r,\alpha_2,\alpha_3)\) is the imaginary Fateev–Litivinov formula [22]: \[\label{eq:FL32formula} \begin{align} &C^{\mathrm{FL}}_\gamma(\kappa\omega_r,\alpha_2,\alpha_3)\\&=\left(l\left(\frac{\gamma^2}{2}\right)\left(\frac{\gamma}{\sqrt{2}}\right)^{2-\gamma^2}\right)^{\frac{\sqrt{2}}{\gamma}\langle 2Q-\kappa\omega_r-\alpha_2-\alpha_3,\rho \rangle}\frac{\Upsilon(\gamma)^r\Upsilon(\kappa)\prod_{e>0}\Upsilon\left(\left\langle Q-\alpha_2,e\right\rangle\right)\Upsilon\left(\left\langle Q-\alpha_3,e\right\rangle\right)}{\prod_{i,j=1}^{r+1}\Upsilon\left(\frac{\kappa}{r+1}+\langle\alpha_2-Q,h_i \rangle+\langle\alpha_3-Q,h_j\rangle\right)}, \end{align}\tag{55}\] with \(\prod_{e>0}\) the product over all positive roots and the weights of the first fundamental representation \(h_1=\omega_1,\) \(h_k=\omega_1-e_1-\cdots-e_{k-1}\) of highest weight \(\omega_r.\)

We provide a proof in Appendix 7.

6 Segal’s gluing axioms↩︎

6.1 Hilbert space of boundary states↩︎

We fix an orthonormal basis \((\varepsilon_j)_{j=1,\ldots,r}\) of \(\mathfrak{a}\). Let \(\mathbb{T}:=\{z\in\mathbb{C}:|z|=1\}\) denote the unit circle A generic \(\mathfrak{a}\)-valued Fourier series on \(\mathbb{T}\) can be written as \[\widetilde{\varphi}(\theta)=c+\varphi(\theta), \quad \varphi(\theta)=\sum_{n\in\mathbb{Z}^*}\sum^r_{j=1}\varphi_{n,j} e^{\boldsymbol{\mathrm{i}}n\theta}\varepsilon_j,\] with constant mode \(c=\sum^r_{j=1}c_j\varepsilon_j\in\mathfrak{a}\) and Fourier coefficients \((\varphi_{n,j})_{n\in\mathbb{Z}^*,j=1,\ldots,r}\) parametrized by \(\varphi_{n,j}=\frac{x_{n,j}+\boldsymbol{\mathrm{i}}y_{n,j}}{2\sqrt n},\,\varphi_{-n,j}=\frac{x_{n,j}-\boldsymbol{\mathrm{i}}y_{n,j}}{2\sqrt n}\) for some \(x_{n,j},y_{n,j}\in\mathbb{R}\).

For \(s\in\mathbb{R}\), we define \[H^s(\mathbb{T};\mathfrak{a}) := \left\{ \varphi=\sum_{n\in\mathbb{Z}}\varphi_n e^{\boldsymbol{\mathrm{i}}n\theta}: \|\varphi\|_{H^s(\mathbb{T};\mathfrak{a})}^2 := \sum_{n\in\mathbb{Z}}\sum^r_{j=1} (1+|n|)^{2s}|\varphi_{n,j}|^2<\infty \right\}.\] If the variables \(x_{n,\ell},y_{n,\ell}\) are independent standard real Gaussian random variables, then the corresponding random Fourier series belongs almost surely to \(H^s(\mathbb{T};\mathfrak{a})\) for every \(s<0\).

We next incorporate the compactification. Recall that the target torus is \(\mathbb{T}(\gamma)=\mathfrak{a}/(2\pi\Lambda),\) with \(\Lambda=\gamma^{-1}\bigoplus_{i=1}^r\mathbb{Z}\omega_i^\vee.\) We define the equivariant Sobolev space \[H^s_\Lambda(\mathbb{R};\mathfrak{a}) := \left\{ u\in H^s_{\mathrm{loc}}(\mathbb{R};\mathfrak{a}): u(\theta+2\pi n)-u(\theta)\in 2\pi\Lambda \quad\text{for all }n\in\mathbb{Z} \right\}.\] Similarly to the discussion in Section 2.5, we have the identification \[\Lambda\times H^s(\mathbb{T};\mathfrak{a})\rightarrow H^s_{\Lambda}(\mathbb{R};\mathfrak{a}),\quad (\lambda,\widetilde{\varphi})\mapsto\pi^*\widetilde{\varphi}(\theta)+\lambda\theta.\]

Let \(\Omega_{\mathbb{T}}:=(\mathbb{R}^{2r})^{\mathbb{N}^*}\), equipped with the cylinder sigma-algebra \(\Sigma_{\mathbb{T}}\), and let \(\mathbb{P}_{\mathbb{T}}\) be the probability measure \[\mathbb{P}_{\mathbb{T}} := \bigotimes_{\substack{n\ge1,\\ j=1,\ldots,r}} \left[ \frac{1}{2\pi} e^{-\frac{1}{2}\left(x_{n,j}^2+y_{n,j}^2\right)} \mathrm{d}x_{n,j}\mathrm{d}y_{n,j} \right].\] The push-forward of \(\mathbb{P}_{\mathbb{T}}\) by \[(x_{n,j},y_{n,j})_{\substack{n\ge1},j=1,\ldots,r}\longmapsto \varphi(\theta) = \sum_{n>0}\sum_{j=1}^r \frac{x_{n,j}+\boldsymbol{\mathrm{i}}y_{n,j}}{2\sqrt n}e^{\boldsymbol{\mathrm{i}}n\theta}\varepsilon_j + \frac{x_{n,j}-\boldsymbol{\mathrm{i}}y_{n,j}}{2\sqrt n}e^{-\boldsymbol{\mathrm{i}}n\theta}\varepsilon_j\] defines a measure on \(H^s(\mathbb{T};\mathfrak{a})\), still denoted \(\mathbb{P}_{\mathbb{T}}\), which is supported on \(H^s(\mathbb{T};\mathfrak{a})\) for any \(s<0,\) i.e., \(\mathbb{P}(\varphi\in H^s(\mathbb{T};\mathfrak{a}))=1\)

We equip \(\Lambda\) with the discrete sigma-algebra and the counting measure \(\mu_\Lambda:=\sum_{\lambda\in\Lambda}\delta_\lambda,\) and \(\mathbb{T}(\gamma)=\mathfrak{a}/(2\pi\Lambda)\) with the Lebesgue measure induced by the Euclidean structure on \(\mathfrak{a}\). We then define the boundary Hilbert space by \[\mathcal{H}:=\mathcal{H}_{\mathfrak{g},\gamma} := L^2\!\left( \mathbb{T}(\gamma)\times\Lambda\times\Omega_{\mathbb{T}}, \, \mathrm{d}c\otimes\mu_\Lambda\otimes\mathbb{P}_{\mathbb{T}} \right).\]

Equivalently, the map \[(c,\lambda,\varphi) \longmapsto u_{\lambda,c,\varphi}(\theta) := c+\lambda\theta+\varphi(\theta)\] pushes forward \(\mathrm{d}c\otimes\mu_\Lambda\otimes\mathbb{P}_{\mathbb{T}}^{\mathfrak{a}}\) to a measure, denoted \(\mu_0\), on \(H^s_\Lambda(\mathbb{R};\mathfrak{a})\). This measure is invariant under global translations \(\tau_\ell:u\longmapsto u+2\pi\lambda,\, \lambda\in\Lambda,\) and therefore descends to the quotient \(H^s_\Lambda(\mathbb{R};\mathfrak{a})/(2\pi\Lambda).\) With this notation we may equivalently write \[\mathcal{H}\simeq L^2\left(H^s_\Lambda(\mathbb{R};\mathfrak{a})/(2\pi\Lambda),\mu_0\right).\] Concretely, if \(F:H^s_\Lambda(\mathbb{R};\mathfrak{a})\to\mathbb{C}\) is measurable and invariant under global translations by an element of the lattice \(\Lambda\), then \[\label{eq:dmu0} \int_{H^s_\Lambda(\mathbb{R};\mathfrak{a})/(2\pi\Lambda)} F(u)\,\mathrm{d}\mu_0(u) = \int_{\mathbb{T}(\gamma)} \sum_{\lambda\in\Lambda} \mathbb{E} \left[ F\bigl(c+\lambda\theta+\varphi(\theta)\bigr) \right] \mathrm{d}c.\tag{56}\]

6.2 Dirichlet-to-Neumann map↩︎

Let \((\Sigma,g)\) be a compact Riemann surface with analytic boundary \(\partial\Sigma=\bigsqcup_{j=1}^{\mathbb{b}}\partial_j\Sigma\). We assume that each \(\partial_j\Sigma\) is equipped with an analytic parametrization \(\zeta_j:\mathbb{T}\longrightarrow \partial_j\Sigma,\) and that the metric \(g\) is chosen so that each boundary component has length \(2\pi\). We denote by \(\mathrm{d}\ell_g\) the Riemannian measure induced by \(g\) on \(\partial\Sigma\).

Boundary data will be written as \(\widetilde{\boldsymbol{\varphi}}=(\widetilde{\varphi}_1,\ldots,\widetilde{\varphi}_{\mathbb{b}}) \in H^s(\mathbb{T};\mathfrak{a})^{\mathbb{b}}.\) We use the following normalized distributional pairing between \(H^s(\mathbb{T};\mathfrak{a})^{\mathfrak b}\) and \(H^{-s}(\mathbb{T};\mathfrak{a})^{\mathfrak b}\): \[\langle \boldsymbol{\varphi},\boldsymbol{\psi}\rangle_{\partial} := \frac{1}{2\pi} \sum_{j=1}^{\mathfrak b} \int_0^{2\pi} \langle\varphi_j(e^{\boldsymbol{\mathrm{i}}\theta}),\psi_j(e^{\boldsymbol{\mathrm{i}}\theta})\rangle \,\mathrm{d}\theta\] whenever the integral expression makes sense.

For smooth boundary data \(\widetilde{\boldsymbol{\varphi}}\in C^\infty(\mathbb{T};\mathfrak{a})^{\mathfrak b}\), we denote by \(P_\Sigma\widetilde{\boldsymbol{\varphi}}\) its harmonic extension to \(\Sigma\), namely the unique smooth \(\mathfrak{a}\)-valued function on \(\Sigma\) satisfying \[\Delta_g P_\Sigma\widetilde{\boldsymbol{\varphi}}=0 \quad\text{in }\Sigma, \qquad P_\Sigma\widetilde{\boldsymbol{\varphi}}|_{\partial_j\Sigma} = \widetilde{\varphi}_j\circ \zeta_j^{-1}, \quad j=1,\ldots,\mathbb{b}.\] For general \(\widetilde{\boldsymbol{\varphi}}\in H^s(\mathbb{T};\mathfrak{a})^{\mathbb{b}}\), the harmonic extension \(P_\Sigma\widetilde{\boldsymbol{\varphi}}\) is understood weakly. More precisely, for every \(u\in C^\infty(\mathbb{T};\mathfrak{a})\), \[\lim_{r\to1^-} \int_0^{2\pi} \langle P_\Sigma\boldsymbol{\varphi}(\zeta_j(re^{\boldsymbol{\mathrm{i}}\theta})),u(e^{\boldsymbol{\mathrm{i}}\theta})\rangle \,\mathrm{d}\theta = 2\pi\langle \varphi_j,u\rangle .\]

For later use, we define the Dirichlet-to-Neumann operator \(\mathbf{D}_\Sigma: C^\infty(\mathbb{T};\mathfrak{a})^{\mathbb{b}} \rightarrow C^\infty(\mathbb{T};\mathfrak{a})^{\mathbb{b}}\) by \[\mathbf{D}_\Sigma\widetilde{\boldsymbol{\varphi}} := \left( -\partial_\nu P_\Sigma\widetilde{\boldsymbol{\varphi}}|_{\partial_j\Sigma}\circ\zeta_j \right)_{j=1,\ldots,\mathbb{b}},\] where \(\nu\) denotes the inward-pointing unit normal vector field along \(\partial\Sigma\). Equivalently, if \(\widetilde{\boldsymbol{\varphi}}=\sum_{\ell=1}^r\widetilde{\boldsymbol{\varphi}}^{(\ell)}\varepsilon_\ell,\) then \(\mathbf{D}_\Sigma\widetilde{\boldsymbol{\varphi}} = \sum_{\ell=1}^r \mathbf{D}_\Sigma^{\mathrm{scal}}\widetilde{\boldsymbol{\varphi}}^{(\ell)}\,\varepsilon_\ell,\) where \(\mathbf{D}_\Sigma^{\mathrm{scal}}\) is the usual scalar Dirichlet-to-Neumann map. In particular, \(\mathbf{D}_\Sigma\) is symmetric and non-negative with kernel given by \(\ker \mathbf{D}_\Sigma = \left\{ (c,\ldots,c):c\in\mathfrak{a} \right\}.\) Indeed, Green’s formula gives \[\label{eq:DN-energy-identity} \int_\Sigma \langle\mathrm{d}P_\Sigma\widetilde{\boldsymbol{\varphi}},\mathrm{d}P_\Sigma\widetilde{\boldsymbol{\varphi}}\rangle_g \,\mathrm{d}\mathrm{v}_g = 2\pi \langle \widetilde{\boldsymbol{\varphi}},\mathbf{D}_\Sigma\widetilde{\boldsymbol{\varphi}}\rangle.\tag{57}\]

We shall also need a Dirichlet-to-Neumann operator associated with cutting curves in the interior of \(\Sigma\). Let \(\mathcal{C}'=\bigsqcup_{j=1}^{\mathbb{b}'}\mathcal{C}'_j\) be a finite collection of pairwise disjoint analytic simple closed curves contained in the interior of \(\Sigma\), and let \(\zeta'_j:\mathbb{T}\to \mathcal{C}'_j\) be analytic parametrizations. For \(\widetilde{\boldsymbol{\varphi}}=(\widetilde{\varphi}_1,\ldots,\widetilde{\varphi}_{\mathbb{b}'}) \in C^\infty(\mathbb{T};\mathfrak{a})^{\mathbb{b}'},\) we denote by \(P_{\mathcal{C}'}\widetilde{\boldsymbol{\varphi}}\) the harmonic function on \(\Sigma\setminus\mathcal{C}'\) with zero boundary condition on \(\partial\Sigma\) and with trace \[P_{\mathcal{C}'}\widetilde{\boldsymbol{\varphi}}|_{\mathcal{C}'_j} = \widetilde{\varphi}_j\circ(\zeta'_j)^{-1}, \quad j=1,\ldots,\mathbb{b}'.\]

Let \(\nu_-\) and \(\nu_+\) denote the two unit normal vector fields along \(\mathcal{C}'_j\), pointing into the two sides of \(\Sigma\setminus\mathcal{C}'\). We define the interior Dirichlet-to-Neumann operator \(\mathbf{D}_{\Sigma,\mathcal{C}'}: C^\infty(\mathbb{T};\mathfrak{a})^{\mathbb{b}'} \longrightarrow C^\infty(\mathbb{T};\mathfrak{a})^{\mathbb{b}'}\) by the formula \[\label{eq:interior-DN-map} \mathbf{D}_{\Sigma,\mathcal{C}'}\widetilde{\boldsymbol{\varphi}} := -\left( \left(\partial_{\nu_-}P_{\mathcal{C}'}\widetilde{\boldsymbol{\varphi}}|_{\mathcal{C}'_j} + \partial_{\nu_+}P_{\mathcal{C}'}\widetilde{\boldsymbol{\varphi}}|_{\mathcal{C}'_j} \right)\circ \zeta'_j \right)_{j=1,\ldots,\mathbb{b}'}.\tag{58}\] The operator \(\mathbf{D}_{\Sigma,\mathcal{C}'}\) is invertible and its inverse has Schwartz kernel \[\label{eq:DN-inverse-kernel} \mathbf{D}_{\Sigma,\mathcal{C}'}^{-1}(y,y') = \frac{1}{2\pi}G_{g,D}(y,y')\,\mathbb{1}_{\mathfrak{a}}, \qquad y\neq y'\in\mathcal{C}',\tag{59}\] where \(\mathbb{1}_{\mathfrak{a}}\) denotes the identity operator on \(\mathfrak{a}\).

For \(k=\mathbb{b}\) or \(k=\mathbb{b}'\), we introduce another operator \(\mathbf{D}: H^1(\mathbb{T};\mathfrak{a})^k \longrightarrow L^2(\mathbb{T};\mathfrak{a})^k\) by declaring that, for \(\widetilde{\boldsymbol{\varphi}} = (\widetilde{\varphi}_1,\ldots,\widetilde{\varphi}_k) \in C^\infty(\mathbb{T};\mathfrak{a})^k,\) with \[\widetilde{\varphi}_j(\theta) = c_j\varepsilon_\ell+ \sum_{n\neq0}\sum_{\ell=1}^r \varphi_{j,n,\ell}e^{\boldsymbol{\mathrm{i}}n\theta}\varepsilon_\ell,\] one has \[(\mathbf{D}\widetilde{\boldsymbol{\varphi}})_j(\theta) = \sum_{n\neq0}\sum_{\ell=1}^r |n|\varphi_{j,n,\ell}e^{\boldsymbol{\mathrm{i}}n\theta}\varepsilon_\ell .\] Equivalently, we have \[\label{eq:model-DN-quadratic-form} \left\langle \mathbf{D}\widetilde{\boldsymbol{\varphi}}, \widetilde{\boldsymbol{\varphi}} \right\rangle_2 = 2\sum_{j=1}^k\sum_{n>0}\sum_{\ell=1}^r n|\varphi_{j,n,\ell}|^2 = \frac{1}{2} \sum_{j=1}^k\sum_{n>0}\sum_{\ell=1}^r \left( x_{j,n,\ell}^2+y_{j,n,\ell}^2 \right).\tag{60}\] We then define the operators on \(C^\infty(\mathbb{T};\mathfrak{a})^\mathbb{b}\) and \(C^\infty(\mathbb{T};\mathfrak{a})^{\mathbb{b}'}\) \[\label{eq:DN-smoothing-remainders} \widetilde{\mathbf{D}}_{\Sigma}:= \mathbf{D}_{\Sigma}-\mathbf{D} ,\qquad \widetilde{\mathbf{D}}_{\Sigma,\mathcal{C}'}:= \mathbf{D}_{\Sigma,\mathcal{C}'}-2\mathbf{D},\tag{61}\] which are smoothing in the sense that their Schwartz kernels are smooth, and for every \(s,s'\in\mathbb{R}\), they extend to bounded operators \[\widetilde{\mathbf{D}}_{\Sigma}: H^s(\mathbb{T};\mathfrak{a})^{\mathbb{b}} \longrightarrow H^{s'}(\mathbb{T};\mathfrak{a})^{\mathbb{b}},\qquad \widetilde{\mathbf{D}}_{\Sigma,\mathcal{C}'}: H^s(\mathbb{T};\mathfrak{a})^{\mathbb{b}'} \longrightarrow H^{s'}(\mathbb{T};\mathfrak{a})^{\mathbb{b}'},\]respectively.

6.3 Curvature terms on surfaces with boundary↩︎

We extend the regularized curvature terms to compact Riemann surfaces with analytic boundary. The construction is the \(\mathfrak{a}\)-valued analogue of [1]. Since all forms and primitives are \(\mathfrak{a}\)-valued, most statements follow componentwise after passing to the fixed orthonormal basis \((\varepsilon_\ell)_{\ell=1,\ldots,r}\) of \(\mathfrak{a}\).

Let \((\Sigma,g)\) be a compact connected Riemann surface with non-empty analytic boundary \(\partial\Sigma=\bigsqcup_{j=1}^{\mathbb{b}}\partial_j\Sigma,\) and let \(\zeta_j:\mathbb{T}\to\partial_j\Sigma, j=1,\ldots,\mathbb{b},\) be analytic parametrizations. We assume that \(g\) is admissible near the boundary, in particular that each boundary component is geodesic and has length \(2\pi\). We denote by \(p_j:=\zeta_j(1)\in\partial_j\Sigma\) the distinguished boundary point on the \(j\)-th boundary component, associated with the winding charges \(\boldsymbol{\lambda}=(\lambda_1,\ldots,\lambda_{\mathbb{b}})\in\Lambda^{\mathbb{b}}\). Let \(\mathbf{v}=((z_1,v_1),\ldots,(z_{n_{\mathfrak{m}}},v_{n_{\mathfrak{m}}}))\) be unit tangent vectors with pairwise distinct base points, to which we attach magnetic charges \(\mathbf{m}=(m_1,\ldots,m_{n_{\mathfrak{m}}})\in\Lambda^{n_{\mathfrak{m}}}\). We impose the boundary magnetic neutrality condition \[\label{eq:boundary-magnetic-neutrality} \sum_{i=1}^{n_{\mathfrak{m}}}m_i+\sum_{j=1}^{\mathbb{b}}\varsigma_j\lambda_j=0.\tag{62}\]

We fix a canonical geometric basis \(\boldsymbol{\sigma}=(\sigma_1,\ldots,\sigma_{2\mathbb{g}+\mathbb{b}-1})\) of the relative homology \(H_1(\Sigma,\partial\Sigma),\) consisting of \(2\mathbb{g}\) interior cycles \((\sigma_1,\ldots,\sigma_{2\mathbb{g}})=(a_1,b_1,\ldots,a_{\mathbb{g}},b_{\mathbb{g}})\) chosen as usual, and \(\mathbb{b}-1\) arcs \((\sigma_{2\mathbb{g}+1},\ldots,\sigma_{2\mathbb{g}+\mathbb{b}-1})=(d_1,\ldots,d_{\mathbb{b}-1})\) with endpoints on the boundary and no intersection with \(\cup_j(a_j\cup b_j)\). We require each \(d_j\) to have endpoints \(p_j,\,p_{j+1}\) and make an angle \(\pi/2\) with \(\partial_j\Sigma\) at its endpoints. We recall from Lemma 3 the identification between \(\boldsymbol{\lambda}^c=(\lambda^c_1,\ldots,\lambda^c_{2\mathbb{g}+\mathbb{b}-1})\in\Lambda^{2\mathbb{g}+\mathbb{b}-1}\) and \(\Omega^c_{\boldsymbol{\lambda}^c},\) where \(\Omega^c_{\boldsymbol{\lambda}^c}\) are chosen to be compactly supported in \(\Sigma^\circ\) with \(\int_{\sigma_k}\Omega^c_{\boldsymbol{\lambda}^c}=2\pi\lambda^c_k\).

We recall from Proposition 4 the closed \(\mathfrak{a}\)-valued \(1\)-form \(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}\) with \[(a_1,b_1,\ldots,a_{\mathbb{g}},b_{\mathbb{g}},\partial_1\Sigma,\ldots,\partial_{\mathbb{b}-1}\Sigma)\] the choice of basis of \(H_1(\Sigma\), where \(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}\) is parametrized by \(\boldsymbol{\lambda}=(\lambda_1,\ldots,\lambda_\mathbb{b})\in\Lambda^{\mathbb{b}}\) and satisfies \[\label{eq:nu-boundary-condition} \int_{\partial_j\Sigma}\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}=2\pi\varsigma_j\lambda_j, \qquad\sum^{n_\mathbf{m}}_{j=1}m_j+\sum^{\mathbb{b}}_{j=1}\varsigma_j\lambda_j=0.\tag{63}\] By Lemma 2 and possibly adding an exact \(\mathfrak{a}\)-valued \(1\)-form \(\mathrm{d}f\) with \(\partial_\nu f|_{\partial\Sigma}=0\), we may require the form \(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}\) to satisfy, for all \(j=1,\ldots,\mathbb{b}\), \[\label{nu-relative-period} \int_{\sigma_{2\mathbb{g}+j}}\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}\in 2\pi\Lambda\tag{64}\] For notational convenience, we set \(z_{n_{\mathbf{m}}+j}=p_j\) and \(m_{n_{\mathbf{m}}+j}=\varsigma_j\lambda_j\) and we associate to the marked points \(\mathbf{z}=(z_1,\ldots,z_{n_{\mathbf{m}}+j})\) the magnetic charges \[\mathbf{m}(\boldsymbol{\lambda})=(m_1(\boldsymbol{\lambda}),\ldots,m_{n_{\mathbf{m}}+\mathbb{b}}(\boldsymbol{\lambda})) = (m_1,\ldots,m_{n_{\mathfrak{m}}},\varsigma_1\lambda_1,\ldots,\varsigma_{\mathbb{b}}\lambda_{\mathbb{b}}) \in\Lambda^{n_{\mathfrak{m}}+\mathbb{b}}.\] For \(j>n_{\mathfrak{m}}\), we choose \(v_j\in T_{p_{j-n_{\mathfrak{m}}}}\Sigma\) to be the unit normal vector to the boundary component at \(p_{j-n_{\mathfrak{m}}}\), with the sign chosen consistently with the orientation convention of the boundary component. We also recall the lexicographic order on \(\Lambda=\gamma^{-1}\bigoplus_{i=1}^r\mathbb{Z}\omega_i^\vee\) induced by the ordered basis \((\omega^\vee_1,\ldots,\omega_r^\vee)\).

As in Lemma 10, the form \(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}\) is exact on \(\Sigma\setminus\mathcal{D}_{\mathbf{v},\boldsymbol{\xi}}.\) Consequently, for \(x_0\in\Sigma\setminus\mathcal{D}_{\mathbf{v},\boldsymbol{\xi}}\), the primitive \[I^{\boldsymbol{\xi}}_{x_0}(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}})(x) := \int_{\alpha_{x_0,x}}\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}\] is well defined on \(\Sigma\setminus\mathcal{D}_{\mathbf{v},\boldsymbol{\xi}}\), where \(\alpha_{x_0,x}\) is any smooth path in this complement.

We now define the magnetic curvature counterterm. For an edge \(\xi_p\), choose \(x\in\xi_p((0,1))\). Let \(\alpha_x\) be a positively oriented \(C^1\) simple loop based at \(x\), contained in a small tubular neighborhood of the defect graph, meeting \(\mathcal{D}_{\mathbf{v},\boldsymbol{\xi}}\) only at \(x\), and starting from the left face of \(\xi_p\). We set \[\label{eq:boundary-kappa-def} \kappa(\xi_p) := \int_{\alpha_x}\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}.\tag{65}\] Equivalently, \[\kappa(\xi_p) = 2\pi\sum_{z_j\in D_{\alpha_x}}m_j(\boldsymbol{\lambda}),\] where \(D_{\alpha_x}\) is the disk bounded by \(\alpha_x\), with the convention that boundary marked points \(p_j\) carry charge \(\varsigma_j\lambda_j\).

We also need the regularized curvature term associated with relative cohomology. Let \(\Sigma_{\boldsymbol{\sigma}}\) be the surface cut along the curves of the basis, on which we define \[I^{\boldsymbol{\sigma}}_{x_0}(\Omega^c_{\boldsymbol{\lambda}^c})(x) := \int_{\alpha_{x_0,x}}\Omega^c_{\boldsymbol{\lambda}^c}.\]

We stress that there is no extra boundary term since the boundary components are geodesic for admissible metrics and \(\Omega^c_{\boldsymbol{\lambda}^c}\) has compact support in \(\Sigma^\circ\).

We collect some properties of the curvature terms, the proofs of which follow from the corresponding lemmas in [1].

We next record the additivity properties of the regularized curvature terms under gluing. These are the \(\mathfrak{a}\)-valued analogues of [1]. Let \((\Sigma_i,g_i)\), \(i=1,2\), be compact Riemann surfaces with admissible metrics, with \(\mathbb{b}_i>0\) analytic boundary components and genera \(\mathbb{g}_i\). We consider the geometric data as above, decorated with a superscript or subscript \(i=1,2.\)

We choose the two defect graphs in the following compatible way. There is exactly one arc \(\xi^1_{j_0}\) in \(\boldsymbol{\xi}_1\) having \(p_{1,\mathbb{b}_1}=\zeta_{1,\mathbb{b}_1}(1)\) as endpoint, and exactly one arc \(\xi^2_{j_0'}\) in \(\boldsymbol{\xi}_2\) having \(p_{2,\mathbb{b}_2}=\zeta_{2,\mathbb{b}_2}(1)\) as endpoint. We assume that \(p_{1,\mathbb{b}_1}\) is glued to \(p_{2,\mathbb{b}_2}\), and that the tangent vectors of \(\xi^1_{j_0}\) and \(\xi^2_{j_0'}\) match under the gluing, namely one is inward normal and the other is outward normal. We also impose the condition \(\lambda_{1,\mathbb{b}_1}=\lambda_{2,\mathbb{b}_2}.\)

Let \(\boldsymbol{\lambda}_i^-:=(\lambda_{i,1},\ldots,\lambda_{i,\mathbb{b}_i-1})\) and \(\boldsymbol{\lambda}:=(\boldsymbol{\lambda}_1^-,\boldsymbol{\lambda}_2^-).\) We obtain a defect graph \(\boldsymbol{\xi}\) on the glued surface \(\Sigma\) by keeping all arcs of \(\boldsymbol{\xi}_1\) except \(\xi^1_{j_0}\), all arcs of \(\boldsymbol{\xi}_2\) except \(\xi^2_{j_0'}\), and adding the arc \(\xi\) obtained by gluing \(\xi^1_{j_0}\) and \(\xi^2_{j_0'}\), with the induced orientation. We also glue the magnetic forms \(\nu^{1}_{\mathbf{z}_1,\mathbf{m}_1,\boldsymbol{\lambda}_1},\,\) \(\nu^{2}_{\mathbf{z}_2,\mathbf{m}_2,\boldsymbol{\lambda}_2}\) to obtain the form \(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}\) on \(\Sigma\), where \(\mathbf{z}=(\mathbf{z}_1,\mathbf{z}_2),\, \mathbf{m}=(\mathbf{m}_1,\mathbf{m}_2).\)

Finally, since the forms \(\Omega^{1,c}_{\boldsymbol{\lambda}_i^c}\) are compactly supported in the interiors of \(\Sigma_i\), they glue to a compactly supported relative cohomology representative on \(\Sigma\), denoted by \(\Omega^c_{\boldsymbol{\lambda}^c} = \Omega^{1,c}_{\boldsymbol{\lambda}_1^c} + \Omega^{2,c}_{\boldsymbol{\lambda}_2^c}.\)

We also need the corresponding self-gluing statement. Let \((\Sigma,g,\boldsymbol{\zeta})\) be a compact Riemann surface with admissible metric, genus \(\mathbb{g}\), and \(\mathbb{b}\ge2\) analytic boundary components. Assume that \(\partial_{\mathbb{b}-1}\Sigma\) is outgoing and that \(\partial_{\mathbb{b}}\Sigma\) is incoming. Let \(\Sigma^\#\) be the surface obtained by identifying \(\partial_{\mathbb{b}-1}\Sigma\) and \(\partial_{\mathbb{b}}\Sigma\) through the boundary parametrizations.

We consider marked tangent vectors \(\mathbf{v}=((z_1,v_1),\ldots,(z_{n_{\mathfrak{m}}},v_{n_{\mathfrak{m}}}))\in(T\Sigma)^{n_{\mathbf{m}}},\) magnetic charges \(\mathbf{m}=(m_1,\ldots,m_{n_{\mathfrak{m}}})\in\Lambda^{n_{\mathfrak{m}}},\) boundary winding data \(\boldsymbol{\lambda}=(\lambda_1,\ldots,\lambda_{\mathbb{b}})\in\Lambda^{\mathbb{b}},\) and the magnetic form \(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}.\) The signs are again chosen so that outgoing boundary components have \(\varsigma=-1\) and incoming ones have \(\varsigma=+1\). We assume that \(\lambda_{\mathbb{b}-1}=\lambda_{\mathbb{b}}\) and write \(\boldsymbol{\lambda}=(\boldsymbol{\lambda}^-,\lambda_{\mathbb{b}},\lambda_{\mathbb{b}}), \, \boldsymbol{\lambda}^-=(\lambda_1,\ldots,\lambda_{\mathbb{b}-2}).\)

We choose the defect graph \(\boldsymbol{\xi}\) on \(\Sigma\) as follows. There is exactly one arc \(\xi_{j_0}\) having \(p_{\mathbb{b}}=\zeta_{\mathbb{b}}(1)\) as endpoint, and the other endpoint of this arc is \(p_{\mathbb{b}-1}=\zeta_{\mathbb{b}-1}(1)\). This arc is oriented according to the tangent vectors at \(p_{\mathbb{b}}\) and \(p_{\mathbb{b}-1}\), one of which is inward normal and the other outward normal. If \(\mathbb{b}+n_{\mathfrak{m}}>2\), we also require that there are exactly two arcs having \(p_{\mathbb{b}-1}\) as endpoint; one of them is \(\xi_{j_0}\), and we denote the other one by \(\xi_{j_0'}.\) After self-gluing, we obtain a defect graph \(\boldsymbol{\xi}^\#\) on \(\Sigma^\#\) by keeping all arcs of \(\boldsymbol{\xi}\) and removing the arcs having \(p_{\mathbb{b}-1}\) or \(p_{\mathbb{b}}\) as endpoint. Finally, one decomposes \(\nu_{\mathbf{z},\mathbf{m}\boldsymbol{\lambda}}\) before gluing as \[\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}} = \nu_{\mathbf{z},\mathbf{m},(\boldsymbol{\lambda}^-,0,0)} + \nu_{\mathbf{z},0,(0,\lambda_{\mathbb{b}},\lambda_{\mathbb{b}})},\] where both terms make sense by [1].

6.4 Amplitudes: definition and gluing↩︎

We now attach to a compact Riemann surface \((\Sigma,g)\), with or without boundary, an amplitude. In the closed case this is nothing but the correlation functional with electro-magnetic operator defined in Theorem 10. When \(\partial\Sigma\neq\emptyset\), the amplitude becomes a functional of the boundary field and will be viewed as an element of the boundary Hilbert space \(\mathcal{H}^{\otimes\mathbb{b}}\) introduced in Section 6.1.

Boundary data will be encoded by \[\widetilde{\boldsymbol{\varphi}}_{\boldsymbol{\lambda}} = (\widetilde{\varphi}_{\lambda_1,1},\ldots,\widetilde{\varphi}_{\lambda_{\mathbb{b}},\mathbb{b}}) \in \bigl(H^s_\Lambda(\mathbb{R};\mathfrak{a})\bigr)^{\mathbb{b}}, \qquad \widetilde{\varphi}_{\lambda_j,j}(\theta)=\lambda_j\theta+\widetilde{\varphi}_j(\theta),\] where \(\widetilde{\boldsymbol{\varphi}}=(\widetilde{\varphi}_1,\ldots,\widetilde{\varphi}_{\mathbb{b}}) \in H^s(\mathbb{T};\mathfrak{a})^{\mathbb{b}}\) is the single-valued part. Let \(P\widetilde{\boldsymbol{\varphi}}\) be the harmonic extension of the single-valued boundary field (Section 6.2). The winding data \(\boldsymbol{\lambda}\) is inserted through the closed \(\mathfrak{a}\)-valued \(1\)-form \(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}\) (defined in Proposition 4), together with a choice of defect graph \(\mathcal{D}_{\mathbf{v},\boldsymbol{\xi}}\) and primitive \(I^{\boldsymbol{\xi}}_{x_0}(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}})\). We recall that \(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}\) does not belong to \(L^2(\Sigma;T^*\Sigma\otimes\mathfrak{a})\). For this reason we systematically use its regularised \(L^2\)-norm \(\|\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}\|_{g,0}\). By the same convention, whenever \(\Omega\) is a smooth \(\mathfrak{a}\)-valued \(1\)-form on \(\Sigma\), we extend the notation and set \[\|\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}+\Omega\|_{g,0}^2 := \|\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}\|_{g,0}^2 +2\langle \nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}},\Omega\rangle_2 +\|\Omega\|_{g,2}^2,\] so that \(\|\cdot\|_{g,0}\) behaves formally as an \(L^2\)-norm on the affine space \(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}+C^\infty(\Sigma;T^*\Sigma\otimes\mathfrak{a})\).

We also need a class of test functionals suited to GFF on a surface with boundary and to the non-uniqueness of the topological decomposition of equivariant fields. Fix \(s<0\). If \(u\in H^s_\Lambda(\widetilde{\Sigma}_{\mathbf{z}};\mathfrak{a})\) is an equivariant distribution on the universal cover of \(\Sigma_{\mathbf{z}}:=\Sigma\setminus\{z_1,\ldots,z_{n_{\mathfrak{m}}}\}\), its monodromies are recorded by the winding vector \(\boldsymbol{\lambda}\) along the boundary components, the magnetic charges \(\mathbf{m}\) around the punctures, and an auxiliary choice of interior cycle data \(\boldsymbol{\lambda}^c\in\Lambda^{2\mathbb{g}+\mathbb{b}-1}\). Choosing \(\boldsymbol{\lambda}^c\) (non-uniquely) and a representative of the corresponding cohomology class, one can write \(u\) in the form \[\label{eq:boundary-decomposition-u} u = \pi^*f + I^{\boldsymbol{\sigma}}_{x_0}(\Omega_{\boldsymbol{\lambda}^c}) + I^{\boldsymbol{\xi}}_{x_0}(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}),\tag{71}\] where \(f\in H^s(\Sigma_{\mathbf{z}};\mathfrak{a})\) is a single-valued Sobolev distribution and \(\pi\) is the covering projection. This decomposition is not canonical: changing the representative within the same cohomology class amounts to replacing \(\Omega_{\boldsymbol{\lambda}^c}+\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}\) by \(\Omega_{\boldsymbol{\lambda}^c}+\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}+ \mathrm{d}h\), for some smooth \(\mathfrak{a}\)-valued function \(h\) on \(\Sigma\) which is locally constant near \(\partial\Sigma\).

We denote by \(\mathcal{E}^{\mathbf{m}}_{\Lambda}(\Sigma)\) the space of functionals \(F\) defined on \(H^s_\Lambda(\widetilde{\Sigma}_{\mathbf{z}};\mathfrak{a})\) with the following property. Fix a connected component of \(H^s_\Lambda(\widetilde{\Sigma}_{\mathbf{z}};\mathfrak{a})\), equivalently a fixed (relative) cohomology class determined by the monodromy data \((\boldsymbol{\lambda}^c,\boldsymbol{\lambda},\mathbf{m})\). Choose a decomposition 71 for \(u\) on this component. We require that \(F\) can be written as \[\label{eq:def-test-functional-boundary} F(u) = P\!\big(\langle f,g_1\rangle,\ldots,\langle f,g_M\rangle\big) G\left( e^{\boldsymbol{\mathrm{i}}\big\langle q,I^{\boldsymbol{\sigma}}_{x_0}(\Omega_{\boldsymbol{\lambda}^c})+ I^{\boldsymbol{\xi}}_{x_0}(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}})\big\rangle} \right),\tag{72}\] where \(q\in\Lambda^*,\) \(g_1,\ldots,g_M\in H^{-s}(\Sigma;\mathfrak{a})\), \(P\) is a polynomial (allowed to depend on \((\boldsymbol{\lambda}^c,\boldsymbol{\lambda},\mathbf{m})\)), and \(G\) is bounded and continuous on \(C^0(\Sigma;\mathbb{S}^1)\) (again allowed to depend on \((\boldsymbol{\lambda}^c,\boldsymbol{\lambda},\mathbf{m})\)).

We stress that \(\mathcal{E}^{\mathbf{m}}_{\Lambda}(\Sigma)\) does not depend on the auxiliary choices entering 71 . Indeed, changing the representative of the cohomology class amounts to replacing \(\Omega_{\boldsymbol{\lambda}^c}+\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}\) by \(\Omega_{\boldsymbol{\lambda}^c}+\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}+\mathrm{d}h\), where \(h\) is a smooth \(\mathfrak{a}\)-valued function on \(\Sigma\), locally constant near \(\partial\Sigma\). Correspondingly, the same \(u\) admits a decomposition with the shifted single-valued part \(f' = f-h\). Since \(h\) is smooth, the pairings satisfy \(\langle f',g_j\rangle=\langle f,g_j\rangle-\langle h,g_j\rangle\), hence the polynomial factor in 72 can be absorbed into a (possibly different) polynomial \(P'\). Moreover, the primitive changes by the addition of \(h-h(x_0)\), so the \(\mathbb{S}^1\)-valued function inside \(G\) is multiplied by the continuous \(\mathbb{S}^1\)-valued function \(\exp(i\langle q,\,h-h(x_0)\rangle)\); composing \(G\) with this multiplication yields another bounded continuous functional \(G'\). Therefore the existence of a representation of the form 72 is invariant under changing representatives, and the definition of \(\mathcal{E}^{\mathbf{m}}_{\Lambda}(\Sigma)\) is well posed.

For \(p\ge 1\), we define the seminorm on \(\mathcal{E}^{\mathbf{m}}_\Lambda(\Sigma;\mathfrak{a})\) \[\label{eq:boundary-seminorm-test-functionals} \begin{align} \|F\|_{\mathcal{L}^{\infty,p}_{\boldsymbol{\alpha},\mathbf{m}}} :={}& \sup_{\boldsymbol{\lambda}^c\in\Lambda^{2\mathbb{g}+\mathbb{b}-1}} \left( \mathbb{E}\!\Bigg[ e^{ -\frac{1}{2\pi}\big\langle \mathrm{d}X_{g,D},\, \Omega_{\boldsymbol{\lambda}^c}^c+\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}\big\rangle_2 -\frac{1}{4\pi}\big\|(1-\Pi_1^c)\Omega_{\boldsymbol{\lambda}^c}\big\|_2^2} \right.\\ &\left.\times \Big|F\!\big(X_{g,D}+P_\Sigma\widetilde{\boldsymbol{\varphi}}+u_{\mathbf{z}}^0+ I^{\boldsymbol{\sigma}}_{x_0}(\Omega^c_{\boldsymbol{\lambda}^c})+I^{\boldsymbol{\xi}}_{x_0}(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}})\big)\Big|^p \Bigg] \right)^{\!1/p}, \end{align}\tag{73}\] where \(u_{z}^0:=\sum^{n_{\mathbf{m}}}_{j=1}i\alpha_jG_{g,D}(\cdot,z_j)\), and \(\Pi_1^c\) is the \(L^2\)-orthogonal projection onto the space of harmonic \(1\)-forms with relative boundary condition so that \((1-\Pi_1^c)\Omega_{\boldsymbol{\lambda}^c}^c=\mathrm{d}f_{\boldsymbol{\lambda}^c}\).

We claim some properties of the amplitudes, whose proof is postponed to Section 6.5.

We now state the gluing identities for the amplitudes. We first recall the geometric setup. Let \((\Sigma_i,g_i,\mathbf{z}_i,\boldsymbol{\zeta}_i),\,i=1,2,\) be two compact connected Riemann surfaces with admissible metrics and parametrized boundary. We write \(\partial\Sigma_i=\bigsqcup_{j=1}^{\mathbb{b}_i}\partial_j\Sigma_i,\,\) \(\boldsymbol{\zeta}_i=(\zeta_{i,1},\ldots,\zeta_{i,\mathbb{b}_i}),\) where each \(\zeta_{i,j}:\mathbb{T}\to\partial_j\Sigma_i\) is an analytic parametrization. We assume that the last boundary component of \(\Sigma_1\) and that of \(\Sigma_2\) are glued together by the parametrizations, with opposite orientations, that \(\partial_{\mathbb{b}-1}\Sigma_1\) is outgoing, that \(x_0^i\in\partial_{\mathbb{b}_i}\Sigma_i\), and that \(x_0^1=x_0^2\) on the glued surface. The resulting surface is denoted by \(\Sigma=\Sigma_1\#\Sigma_2,\) whose remaining boundary parametrization is \(\boldsymbol{\zeta} = (\zeta_{1,1},\ldots,\zeta_{1,\mathbb{b}_1-1}, \zeta_{2,1},\ldots,\zeta_{2,\mathbb{b}_2-1}).\)

The metric \(g\) on \(\Sigma\) is the metric obtained by gluing \(g_1\) and \(g_2\), and is again assumed admissible. The marked points, tangent vectors, electric charges, and magnetic charges are the union of the data on the two pieces: \[\mathbf{z}=(\mathbf{z}_1,\mathbf{z}_2),\quad \mathbf{v}=(\mathbf{v}_1,\mathbf{v}_2),\quad \boldsymbol{\alpha}=(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2),\quad \mathbf{m}=(\mathbf{m}_1,\mathbf{m}_2).\] We also take test functionals \(F_i\in\mathcal{E}^{\mathbf{m}_i}_{\Lambda}(\Sigma_i;\mathfrak{a})\), and write \(F_1\otimes F_2(\Phi_g)=F(\Phi_g|_{\Sigma_1})F(\Phi_g|_{\Sigma_2})\) for the corresponding functional on the glued surface, evaluated on the two restrictions of the field. We write the boundary data on the glued surface \(\Sigma\) as \((\widetilde{\boldsymbol{\varphi}}^{\boldsymbol{\lambda}_1}_1,\widetilde{\boldsymbol{\varphi}}^{\boldsymbol{\lambda}_2}_2)\in (H^s_\Lambda(\mathbb{R};\mathfrak{a}))^{\mathbb{b}_1-1}\times(H^s_\Lambda(\mathbb{R};\mathfrak{a}))^{\mathbb{b}_2-1}\) so that boundary data on \(\Sigma_i\) before gluing takes the form \((\widetilde{\boldsymbol{\varphi}}_{1}^{\boldsymbol{\lambda}_1},\widetilde{\varphi}^\lambda)\).

We will also need the analogous self-gluing setup. Let \((\Sigma,g,\boldsymbol{\zeta})\) be a compact connected Riemann surface with admissible metric and \(\mathbb{b}\ge2\) parametrized boundary components. We assume that the last two boundary components are glued together, with opposite orientations and compatible parametrizations, and denote the resulting surface by \(\Sigma^\#,\) whose remaining boundary parametrizations are denoted by \(\boldsymbol{\zeta}_\#\). The boundary data on the glued surface \(\Sigma^\sharp\) will be written as \(\widetilde{\boldsymbol{\varphi}}^{\boldsymbol{\lambda}} = (\widetilde{\varphi}^{\lambda_1},\ldots, \widetilde{\varphi}^{\lambda_{\mathbb{b}-2}})\in(H^s_{\Lambda}(\mathbb{R};\mathfrak{a}))^{\mathbb{b}-2}\), and that on the unglued surface \(\Sigma\) will be written as \((\widetilde{\boldsymbol{\varphi}}^{\boldsymbol{\lambda}},\widetilde{\varphi}^{\lambda_{\mathbb{b}-1}},\widetilde{\varphi}^{\lambda_{\mathbb{b}}})\in (H^s_{\Lambda}(\mathbb{R};\mathfrak{a}))^{\mathbb{b}-2}\times H^s_{\Lambda}(\mathbb{R};\mathfrak{a})\times H^s_{\Lambda}(\mathbb{R};\mathfrak{a})\), where \(\widetilde{\varphi}^{\lambda_{\mathbb{b}-1}},\,\widetilde{\varphi}^{\lambda_{\mathbb{b}-1}}\) denote the boundary data on \(\partial_{\mathbb{b}-1}\Sigma,\,\partial_{\mathbb{b}}\Sigma,\) respectively.

6.5 Proof of Propositions 13 and 14↩︎

We start with the case of free compactified boson, namely we restrict to the case where \(\boldsymbol{\alpha}=0,\, \boldsymbol{\mu}=0,\, Q=0.\) In particular, there is no curvature phase and no Toda interaction. The only remaining nontrivial terms are the compactification, the topological sectors, and the magnetic background \(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}\).

For a positive measurable functional \(F\), or more generally for an integrable measurable functional for which the expression below is finite, we set \[\label{eq:free-compactified-boson-amplitude} \begin{align} \mathcal{A}^0_{\Sigma,g,\mathbf{z},\mathbf{m},\boldsymbol{\zeta}} \left(F,\widetilde{\boldsymbol{\varphi}}^{\boldsymbol{\lambda}}\right) :=& \delta_0\!\left(\sum_{j=1}^{n_{\mathfrak m}+\mathbb{b}}m_j(\boldsymbol{\lambda})\right) \sum_{\boldsymbol{\lambda}^c\in\Lambda^{2\mathbb{g}+\mathbb{b}-1}} e^{-\frac{1}{4\pi} \left\|\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}+\Omega^c_{\boldsymbol{\lambda}^c}\right\|_{g,0}^2} Z_{\Sigma,g}\, \mathcal{A}^0_{\Sigma,g}(\widetilde{\boldsymbol{\varphi}}) \\ &\times \mathbb{E}\left[ e^{ -\frac{1}{2\pi} \left\langle \mathrm{d}\mathbf{X}_{g,D}+\mathrm{d}P\widetilde{\boldsymbol{\varphi}}, \nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}+\Omega^c_{\boldsymbol{\lambda}^c} \right\rangle_2 }F(\Phi_g^0) \right], \end{align}\tag{79}\] where \[\Phi_g^0 := \mathbf{X}_{g,D} + P\widetilde{\boldsymbol{\varphi}} + I^{\boldsymbol{\xi}}_{x_0}(\nu_{\mathbf{z},\mathbf{m},\boldsymbol{\lambda}}) + I^{\boldsymbol{\sigma}}_{x_0}(\Omega^c_{\boldsymbol{\lambda}^c}).\] Here \(F\) is assumed to be \(2\pi\Lambda\)-periodic: \(F(\Phi+2\pi\lambda)=F(\Phi), \, \lambda\in\Lambda.\) The amplitude \(\mathcal{A}^0_{\Sigma,g,\mathbf{z},\mathbf{m},\boldsymbol{\zeta}}\) has no dependence on the tangent vectors \(\mathbf{v}\), except through the auxiliary choice of defect graph used to write the primitive; by the defect-graph independence established above, the resulting amplitude is independent of that choice.

We now record the free gluing identities. They are the compactified-boson analogues of Proposition 14, and they are the basic input for the interacting Toda gluing identities.

Now we are in a position to prove Propositions 13 and 14. We begin with the former.

We conclude this section by proving Proposition 14.

7 Proof of Proposition 11↩︎

7.1 The complex twin and meromorphic continuation↩︎

We first introduce a few notations. We denote by \(\Lambda_{\mathbb{C}}\) the collection of pairs \((a,a')\in\mathbb{C}^2\) such that \(a-a'\in\mathbb{Z}\) and write \(z^{a|a'}=z^a\bar z^{a'}\) with the choice of branch such that \((-1)^{a|a'}=(-1)^{a-a'}.\) We then define the Gamma function of the complex field [37] by \[\label{eq:complex32Gamma}\Gamma^\mathbb{C}(a|a'):=\frac{1}{\pi}\int_{\mathbb{C}}z^{a-1|a'-1}e^{2\boldsymbol{\mathrm{i}}\mathrm{Re}z}\mathrm{d}z=\boldsymbol{\mathrm{i}}^{a-a'}\frac{\Gamma(a)}{\Gamma(1-a')}=\frac{\boldsymbol{\mathrm{i}}^{a'-a}}{\pi}\Gamma(a)\Gamma(a')\sin(\pi a').\tag{91}\] This integral converges conditionally if \(0<\frac{1}{2}\mathrm{Re}(a+a')<1\) and admits a meromorphic continuation to the complex lines \(a-a'\in\mathbb{Z}\). Finally, for \(\mathbf{z}=(z_1,\ldots,z_n)\in\mathbb{C}^n\), we let \(D_n(\mathbf{z})=\prod_{1\leq i<j\leq n}(z_i-z_j)\) and \(\mathrm{d}\nu_n(z)=(\pi^n n!)^{-1}\prod_{i=1}^n\mathrm{d}z_i\).

We also record the following identity: for \((\lambda,\lambda')\in\Lambda_{\mathbb{C}}\) satisfying \(-2<\mathrm{Re}(\lambda+\lambda')<0\) and \(a\in\mathbb{C}^\times\), we have \[\label{eq:one-variable-Fourier} \frac{1}{\pi}\int_{\mathbb{C}} z^{\lambda|\lambda'}e^{\,a\bar z-\bar a z}\mathrm{d}z = \Gamma_{\mathbb{C}}(1+\lambda\,|\,1+\lambda')\,a^{-1-\lambda'\,|\,-1-\lambda}.\tag{92}\] This can be obtained from 91 via the change of variable \(w=\bar{a}z\).

To follow the argument in [20], we have to establish the complex twin of [20] since the integrand in 52 is no longer the modulus of, say, \(1-x^{(i)}_a\) due to the nonzero magnetic charge \(m_2.\)

For later applications of Lemma 29, we will need to evaluate both sides of 93 for parameters outside the common absolute-convergence region 94 . We therefore establish a general meromorphic continuation theorem for complex local zeta integrals and then apply it to the present setting.

The argument follows the method of Atiyah [24], namely to reduce the integral, after resolution of singularities, to finitely many monomial Mellin-type integrals. For the resolution input, we use principalization of ideal sheaves in the complex-analytic category [25], [38]. As a consequence, the polynomial case on \(\mathbb{C}^n\) follows immediately, and this will imply the meromorphic continuation of both sides of 93 .

7.2 Proof of Proposition 11↩︎

Let \(\mathcal{I}_{\mathbf{s}}(\sigma,\tau,B_1,B_1',\ldots, B_r,B_r')\) denote the integral \[\begin{align} \int_{\mathbb{C}^N} \prod_{a=1}^{s_r}(x_a^{(r)})^{\sigma|\sigma} \prod_{i=1}^{r}\prod_{a=1}^{s_i}(&1-x_a^{(i)})^{B_i|B_i'} \prod_{i=1}^{r}D_{s_i}(x^{(i)})^{\tau|\tau} \\&\times\prod_{i=1}^{r-1}\prod_{a=1}^{s_i}\prod_{b=1}^{s_{i+1}} (x_b^{(i+1)}-x_a^{(i)})^{-\tau/2|-\tau/2} \prod_{i=1}^{r}\mathrm{d}\nu_{s_i}(x^{(i)}),\end{align}\] where \[B_i=\frac{\gamma}{2}\langle e_i,\alpha_2+m_2\rangle,\quad B'_i=\frac{\gamma}{2}\langle e_i,\alpha_2-m_2\rangle.\] We recall that the Cartan matrix of the lie algebra \(\mathfrak{sl}_{r+1}\) is given by \[A_{ij}=\begin{cases} 2 &\text{if }i=j,\\ -1 &\text{if } |i-j|=1,\\ 0 &\text{otherwise}, \end{cases}\] and so the \(A_r\)-Dotsenko–Fateev integral 52 becomes \[\label{eq:to32be32proved} \begin{align} \prod_{i=1}^r(\pi^{s_i} s_i!)\mathcal{I}_{\mathbf{s}}\left(\frac{\kappa\gamma}{2},\gamma^2, B_1,B'_1,\ldots,B_r,B_r'\right). \end{align}\tag{102}\] It remains to show that the square of the integral in 102 coincides with the product of the imaginary Fateev–Litivinov formula as showcased in ?? .

We follow the argument in [20] with Proposition 29 in place of [20] to first obtain a recurrent relation 116 and then apply the shift equation 54 of the special function \(\Upsilon\) to conclude the proof.

The applications of Lemma 29 below are first carried out in a non-empty auxiliary open subset of the parameter space \((\sigma,\tau,B_1,B_1',\ldots,B_r,B_r')\in \mathbb{C}^2\times\Lambda_{\mathbb{C}}^r,\) where \(\sigma\) and \(\tau\) are kept diagonal, i.e. they correspond to the exponents \(\sigma|\sigma\) and \(\tau|\tau\). On this subset all intermediate integrals are absolutely convergent, and the applications of Lemma 29 give the recurrence relation 116 . By Corollary 2, both sides of this recurrence relation admit componentwise meromorphic continuations to \(\mathbb{C}^2\times\Lambda_{\mathbb{C}}^r\). Hence the recurrence relation, and therefore its iteration, holds meromorphically. Only after the full iterated formula has been obtained do we specialize to the Toda parameters. At exceptional parameter values, the identity is understood by meromorphic continuation.

Although in Steps 1–4 below we set \(\sigma=\frac{\kappa\gamma}{2},\,\tau=\gamma^2\), this is only a notational simplification. The argument should be understood as the preceding meromorphic-continuation argument with \(\sigma\) and \(\tau\) first treated as independent diagonal complex-field parameters. The specialization \(\sigma=\frac{\kappa\gamma}{2}, \,\tau=\gamma^2\) is imposed only after the recurrence and its iteration have been established as meromorphic identities.

7.2.0.1 Step 1: Removing the row \(x^{(1)}\)

We begin by rewriting the Vandermonde factor in the first row. Lemma 29 with \(p=s_1-1,\,q=0,\) and \(\lambda_j=\gamma^2/2-1\) implies that

\[\label{eq:first-row-m0} D_{s_1}(x^{(1)})^{\gamma^2-1|\gamma^2-1} = C_0 \int_{\mathbb{C}^{s_1-1}} D_{s_1-1}(y^{(1)})^{1|1} \prod_{\beta=1}^{s_1-1}\prod_{a=1}^{s_1} (y_\beta^{(1)}-x_a^{(1)})^{\frac{\gamma^2}{2}-1|\frac{\gamma^2}{2}-1} \mathrm{d}\nu_{s_1-1}(y^{(1)}),\tag{103}\] with \[\label{eq:c1} C_0= \frac{ \Gamma^{\mathbb{C}}\!\left(\frac{s_1\gamma^2}{2}\,\Big|\,\frac{s_1\gamma^2}{2}\right) }{ \Gamma^{\mathbb{C}}\!\left(\frac{\gamma^2}{2}\,\Big|\,\frac{\gamma^2}{2}\right)^{s_1} }=\frac{l\left(\frac{s_1\gamma^2}{2}\right)}{l\left(\frac{\gamma^2}{2}\right)^{s_1}},\tag{104}\] where we have used 91 and \(l(x)=\Gamma(x)/\Gamma(1-x)\). We note that the sign in 93 is positive since there is only one exponent \(\lambda_j|\lambda_j'\) that is not of the form \(\lambda|\lambda\), and so the exponent \(\sum_{j=1}^M(j-1)(\lambda_j-\lambda_j')=0\). This argument applies to all the applications of Lemma 29 below.

Using \[D_{s_1}(x^{(1)})^{\gamma^2|\gamma^2}=D_{s_1}(x^{(1)})^{\gamma^2-1|\gamma^2-1}D_{s_1}(x^{(1)})^{1|1}\] and substituting 103 into the integral in 102 , we isolate the remaining \(x^{(1)}\)-integral: \[\label{eq:J1-definition} \begin{align} J_1(y^{(1)},x^{(2)}) := \int_{\mathbb{C}^{s_1}} & D_{s_1}(x^{(1)})^{1|1} \prod_{a=1}^{s_1}(1-x_a^{(1)})^{B_1|B_1'} \\ &\times \prod_{\beta=1}^{s_1-1}\prod_{a=1}^{s_1} (y_\beta^{(1)}-x_a^{(1)})^{\frac{\gamma^2}{2}-1|\frac{\gamma^2}{2}-1} \prod_{a=1}^{s_1}\prod_{b=1}^{s_2} (x_b^{(2)}-x_a^{(1)})^{-\frac{\gamma^2}{2}|-\frac{\gamma^2}{2}} \mathrm{d}\nu_{s_1}(x^{(1)}). \end{align}\tag{105}\]

Now we apply Lemma 29 to \(J_1\), with external points \(1,\;y_1^{(1)},\dots,y_{s_1-1}^{(1)},\;x_1^{(2)},\dots,x_{s_2}^{(2)},\) and exponents \[B_1|B_1', \qquad \underbrace{\frac{\gamma^2}{2}-1|\frac{\gamma^2}{2}-1,\dots,\frac{\gamma^2}{2}-1|\frac{\gamma^2}{2}-1}_{s_1-1\text{ times}}, \qquad \underbrace{-\frac{\gamma^2}{2}|-\frac{\gamma^2}{2},\dots,-\frac{\gamma^2}{2}|-\frac{\gamma^2}{2}}_{s_2\text{ times}}\] to obtain

\[\label{eq:J1} \begin{align} J_1=&C_1 \prod_{\beta=1}^{s_1-1} (1-y_\beta^{(1)})^{B_1+\frac{\gamma^2}{2}|B_1'+\frac{\gamma^2}{2}} D_{s_1-1}(y^{(1)})^{\gamma^2-1|\gamma^2-1} \prod_{a=1}^{s_2} (1-x_a^{(2)})^{1+B_1-\frac{\gamma^2}{2}|1+B_1'-\frac{\gamma^2}{2}} \\ &\times D_{s_2}(x^{(2)})^{1-\gamma^2|1-\gamma^2} \int_{\mathbb{C}^{s_2-1}} \prod_{\delta=1}^{s_2-1} (1-y_\delta^{(2)})^{-1-B_1\,|\,-1-B_1'} D_{s_2-1}(y^{(2)})^{1|1} \\&\prod_{\beta=1}^{s_1-1}\prod_{\delta=1}^{s_2-1} (y_\delta^{(2)}-y_\beta^{(1)})^{-\frac{\gamma^2}{2}|-\frac{\gamma^2}{2}} \times\prod_{\delta=1}^{s_2-1}\prod_{a=1}^{s_2} (x_a^{(2)}-y_\delta^{(2)})^{-1+\frac{\gamma^2}{2}|-1+\frac{\gamma^2}{2}} \mathrm{d}\nu_{s_2-1}(y^{(2)}), \end{align}\tag{106}\] with the constant \(C_1\) given by \[\label{eq:C1} \begin{align} C_1=& \frac{ \Gamma^{\mathbb{C}}(1+B_1\,|\,1+B_1')\, \Gamma^{\mathbb{C}}(\frac{\gamma^2}{2}|\frac{\gamma^2}{2})^{s_1-1}\, \Gamma^{\mathbb{C}}\!\left(1-\frac{\gamma^2}{2}\Big|1-\frac{\gamma^2}{2}\right)^{s_2} }{ \Gamma^{\mathbb{C}}\!\left( s_1+1+B_1+(s_1-1)(\frac{\gamma^2}{2}-1)-\frac{\gamma^2}{2}s_2 \Big|\; s_1+1+B_1'+(s_1-1)(\frac{\gamma^2}{2}-1)-\frac{\gamma^2}{2}s_2 \right) }\\ =&\frac{\Gamma^{\mathbb{C}}(1+B_1\,|\,1+B_1') l\left(\frac{\gamma^2}{2}\right)^{s_1-1}l\left(1-\frac{\gamma^2}{2}\right)^{s_2} }{\Gamma^{\mathbb{C}}\!\left( s_1+1+B_1+(s_1-1)(\frac{\gamma^2}{2}-1)-\frac{\gamma^2}{2}s_2 \Big|\; s_1+1+B_1'+(s_1-1)(\frac{\gamma^2}{2}-1)-\frac{\gamma^2}{2}s_2 \right) } \end{align}\tag{107}\] We note that the term \(D_{s_1-1}(y^{(1)})\) in 103 and 107 are combined to have exponent \(\gamma^2|\gamma^2.\)

7.2.0.2 Step 2: Removing the row \(x^{(2)}\)

We observe that the \(x^{(2)}\)-dependent part becomes \[\begin{align} J_2(y^{(2)},x^{(3)}) := \int_{\mathbb{C}^{s_2}} & D_{s_2}(x^{(2)})^{1|1} \prod_{a=1}^{s_2}(1-x_a^{(2)})^{Q_2|Q_2'} \\ &\times \prod_{\delta=1}^{s_2-1}\prod_{a=1}^{s_2} (x_a^{(2)}-y_\delta^{(2)})^{-1+\frac{\gamma^2}{2}|-1+\frac{\gamma^2}{2}} \prod_{a=1}^{s_2}\prod_{b=1}^{s_3} (x_b^{(3)}-x_a^{(2)})^{-\frac{\gamma^2}{2}|-\frac{\gamma^2}{2}} \mathrm{d}\nu_{s_2}(x^{(2)}), \end{align}\] with \[Q_2:=1+B_1+B_2-\frac{\gamma^2}{2}, \qquad Q_2':=1+B_1'+B_2'-\frac{\gamma^2}{2}.\] Integrating out the second row \(x^{(2)}\) using Lemma 29 yields \[\label{eq:J2} \begin{align} J_2=& C_2 \prod_{a=1}^{s_2-1} (1-y_a^{(2)})^{1+B_1+B_2\,|\,1+B_1'+B_2'} D_{s_2-1}(y^{(2)})^{-1+\gamma^2|-1+\gamma^2} \\ &\times \prod_{b=1}^{s_3} (1-x_b^{(3)})^{2+B_1+B_2-\gamma^2|2+B_1'+B_2'-\gamma^2} D_{s_3}(x^{(3)})^{1-\gamma^2|1-\gamma^2} \\ &\times\int_{\mathbb{C}^{s_3-1}} \prod_{\delta=1}^{s_3-1} (1-y_\delta^{(3)})^{-2-B_1-B_2+\frac{\gamma^2}{2}|-2-B_1'-B_2'+\frac{\gamma^2}{2}} D_{s_3-1}(y^{(3)})^{1|1} \\ &\times \prod_{\eta=1}^{s_2-1}\prod_{\delta=1}^{s_3-1} (y_\delta^{(3)}-y_\eta^{(2)})^{-\frac{\gamma^2}{2}|-\frac{\gamma^2}{2}} \prod_{\delta=1}^{s_3-1}\prod_{a=1}^{s_3} (x_a^{(3)}-y_\delta^{(3)})^{-1+\frac{\gamma^2}{2}|-1+\frac{\gamma^2}{2}} \mathrm{d}\nu_{s_3-1}(y^{(3)}), \end{align}\tag{108}\] with the constant \(C_2\) given by \[\label{eq:C2} C_2= \frac{ \Gamma^{\mathbb{C}}(1+Q_2\,|\,1+Q_2')l\left(\frac{\gamma^2}{2}\right)^{s_2-1} l\left(1-\frac{\gamma^2}{2}\right)^{s_3} }{ \Gamma^{\mathbb{C}}\!\left( s_2+1+Q_2+(s_2-1)(\frac{\gamma^2}{2}-1)-\frac{\gamma^2}{2}s_3 \;\Big|\; s_2+1+Q_2'+(s_2-1)(\frac{\gamma^2}{2}-1)-\frac{\gamma^2}{2}s_3 \right) }.\tag{109}\] We note that the terms \((1-y^{(2)}_a)\) and \(D_{s_2-1}(y^{(2)})\) in 106 and 108 can be combined to have exponents \(B_2|B_2'\) and \(\gamma^2|\gamma^2,\) respectively.

7.2.0.3 Step 3: Removing the row \(x^{k},\,3\leq k\leq r-1\)

We define recursively \[\label{eq:Qk-recursion} Q_{k+1}=1+Q_k+B_{k+1}-\frac{\gamma^2}{2}, \qquad Q_{k+1}'=1+Q_k'+B_{k+1}'-\frac{\gamma^2}{2}, \qquad k\ge 2,\tag{110}\] and thus \[\label{eq:Qk-explicit} Q_k=(k-1)+\sum_{j=1}^k B_j-(k-1)\frac{\gamma^2}{2}, \qquad Q_k'=(k-1)+\sum_{j=1}^k B_j'-(k-1)\frac{\gamma^2}{2}.\tag{111}\] For \(3\le k\le r-1\), the \(x^{(k)}\)-dependent part is \[\label{eq:Jk} \begin{align} J_k(y^{(k)},x^{(k+1)}) := &\int_{\mathbb{C}^{s_k}} D_{s_k}(x^{(k)})^{1|1} \prod_{a=1}^{s_k}(1-x_a^{(k)})^{Q_k|Q_k'} \\ &\times \prod_{\alpha=1}^{s_k-1}\prod_{a=1}^{s_k} (x_a^{(k)}-y_\alpha^{(k)})^{-1+\frac{\gamma^2}{2}|-1+\frac{\gamma^2}{2}} \prod_{a=1}^{s_k}\prod_{b=1}^{s_{k+1}} (x_b^{(k+1)}-x_a^{(k)})^{-\frac{\gamma^2}{2}|-\frac{\gamma^2}{2}} \mathrm{d}\nu_{s_k}(x^{(k)}). \end{align}\tag{112}\]

Applying Lemma 29 to \(J_k\), we obtain a new row \(y^{(k+1)}=(y_1^{(k+1)},\dots,y_{s_{k+1}-1}^{(k+1)}),\) and the constant \[\label{eq:Ck} C_k= \frac{ \Gamma^{\mathbb{C}}(1+Q_k\,|\,1+Q_k')l\left(\frac{\gamma^2}{2}\right)^{s_k-1} l\left(1-\frac{\gamma^2}{2}\right)^{s_{k+1}} }{ \Gamma^{\mathbb{C}}\!\left( s_k+1+Q_k+(s_k-1)(\frac{\gamma^2}{2}-1)-\frac{\gamma^2}{2}s_{k+1} \;\Big|\; s_k+1+Q_k'+(s_k-1)(\frac{\gamma^2}{2}-1)-\frac{\gamma^2}{2}s_{k+1} \right) }.\tag{113}\]

We note that the terms \((1-y^{(k)}_a)\) and \(D_{s_k-1}(y^{(k)})\) have exponents \(B_k|B_k'\) and \(\gamma^2|\gamma^2,\) respectively.

7.2.0.4 Step 4: Removing the row \(x^{(r)}\)

Finally the \(x^{(r)}\)-dependent term takes the form \[\label{eq:Jr} \begin{align} J_r(y^{(r)}) := \int_{\mathbb{C}^{s_r}} D_{s_r}(x^{(r)})^{1|1} &\prod_{a=1}^{s_r}(x_a^{(r)})^{\frac{\kappa\gamma}{2}|\frac{\kappa\gamma}{2}}(1-x_a^{(r)})^{Q_r|Q_r'}\\&\times \prod_{\alpha=1}^{s_r-1}\prod_{a=1}^{s_r} (x_a^{(r)}-y_\alpha^{(r)})^{-1+\frac{\gamma^2}{2}|-1+\frac{\gamma^2}{2}} \mathrm{d}\nu_{s_r}(x^{(r)}). \end{align}\tag{114}\] We apply again Lemma 29 to obtain that \[\begin{align} J_r=C_r\prod_{a=1}^{s_r-1}\left(y_a^{(r)}\right)^{\frac{\kappa\gamma+\gamma^2}{2}|\frac{\kappa\gamma+\gamma^2}{2}}\left(1-y^{(r)}_a\right)^{Q_r+\frac{\gamma^2}{2}|Q_r+\frac{\gamma^2}{2}}D_{s_r-1}(y^{(r)})^{-1+\gamma^2|-1+\gamma^2}, \end{align}\] with the constant \(C_r\) given by \[\label{eq:Kr-explicit} C_r= \frac{ \Gamma^{\mathbb{C}}(1+Q_r\,|\,1+Q_r')l\left(1+\frac{\kappa\gamma}{2}\right) l\left(\frac{\gamma^2}{2}\right)^{s_r-1} }{ \Gamma^{\mathbb{C}}\!\left( s_r+1+\frac{\kappa\gamma}{2}+Q_r+(s_r-1)\!\left(-1+\frac{\gamma^2}{2}\right) \;\Big|\; s_r+1+\frac{\kappa\gamma}{2}+Q_r'+(s_r-1)\!\left(-1+\frac{\gamma^2}{2}\right) \right) }.\tag{115}\]

We combine all the above results to conclude that \[\label{eq:recurrent32relation} \mathcal{I}_{\mathbf{s}}\left(\frac{\kappa\gamma}{2},\gamma^2,B_1,B_1',\ldots,B_r,B_r'\right)=\prod_{k=0}C_k\mathcal{I}_{\mathbf{s}-\mathbf{1}}\left(\frac{\kappa\gamma+\gamma^2}{2},\gamma^2,B_1+\frac{\gamma^2}{2},B_1'+\frac{\gamma^2}{2},\ldots,B_r,B_r'\right),\tag{116}\] with \(\mathbf{s}-\mathbf{1}=(s_1-1,\ldots,s_r-1).\)

7.2.0.5 Step 5: Conclusion

By 91 , one has \[\Gamma^{\mathbb{C}}(a|a')\Gamma^{\mathbb{C}}(a'|a) = \frac{\Gamma(a)}{\Gamma(1-a)} \frac{\Gamma(a')}{\Gamma(1-a')} = l(a)l(a'), \quad (a,a')\in\Lambda_{\mathbb{C}}.\] Let \(\mathcal{R}_{\mathbf{s}}:=C_0C_1\cdots C_r\) be the recurrence factor in 116 . Then writing \(B^+_j=B_j\) and \(B^-_j=B'_j\), we have \(\mathcal{R}_{\mathbf{s}}^2=\mathcal{R}^+_{\mathbf{s}}\mathcal{R}^-_{\mathbf{s}}\), with \[\begin{align} \mathcal{R}_{\mathbf{s}}^{\pm} &= l\left(\frac{s_1\gamma^2}{2}\right) l\left(1+\frac{\kappa\gamma}{2}\right) l\left(\frac{\gamma^2}{2}\right)^{-r} \\ &\quad\times \prod_{k=1}^{r-1} \frac{ l\left( k+\sum_{j=1}^{k}B_j^{\pm} -(k-1)\frac{\gamma^2}{2} \right) }{ l\left( k+1+\sum_{j=1}^{k}B_j^\pm +\frac{\gamma^2}{2}(s_k-s_{k+1}-k) \right) } \\ &\quad\times \frac{ l\left( r+\sum_{j=1}^{r}B_j^\pm -(r-1)\frac{\gamma^2}{2} \right) }{ l\left( r+1+\frac{\kappa\gamma}{2} +\sum_{j=1}^{r}B_j^\pm +\frac{\gamma^2}{2}(s_r-r) \right) }, \end{align}\] where we have used \[l\left(1-\frac{\gamma^2}{2}\right) = l\left(\frac{\gamma^2}{2}\right)^{-1}.\]

We now rewrite the sums \(\sum_{j=1}^{k}B^\pm_j\) in terms of the weights of the fundamental representation of highest weight \(\omega_r\). Let \(h_1,\ldots,h_{r+1}\) be such weights, ordered so that \(h_{k+1}-h_k=e_k,\,1\le k\le r.\) Then \(h_{k+1}-h_1=\sum_{j=1}^{k}e_j,\) and hence \(\sum_{j=1}^{k}B^\pm_j=\left\langle h_{k+1}-h_1,\alpha_2\pm m_2\right\rangle.\) This shows that \[\label{eq:Rbs-pm-h} \begin{align} \mathcal{R}_{\mathbf{s}}^{\pm} &= l\left(\frac{s_1\gamma^2}{2}\right)\, l\left(1+\frac{\kappa\gamma}{2}\right)\, l\left(\frac{\gamma^2}{2}\right)^{-r} \\ &\quad\times \prod_{k=1}^{r-1} \frac{ l\!\left( k+\bigl\langle h_{k+1}-h_1,\alpha_2\pm m_2\bigr\rangle-(k-1)\frac{\gamma^2}{2} \right) }{ l\!\left( k+1+\bigl\langle h_{k+1}-h_1,\alpha_2\pm m_2\bigr\rangle+\frac{\gamma^2}{2}(s_k-s_{k+1}-k) \right) } \\ &\quad\times \frac{ l\!\left( r+\bigl\langle h_{r+1}-h_1,\alpha_2\pm m_2\bigr\rangle-(r-1)\frac{\gamma^2}{2} \right) }{ l\!\left( r+1+\frac{\kappa\gamma}{2}+\bigl\langle h_{r+1}-h_1,\alpha_2\pm m_2\bigr\rangle+\frac{\gamma^2}{2}(s_r-r) \right) }. \end{align}\tag{117}\]

We recall the special function \(\Upsilon\) defined in 53 and the shift relation 54 : \[\label{eq:Upsilon-shift-proof-detail} \Upsilon(z+\chi) = l\!\left(\frac{\chi z}{2}\right) \left(\frac{\chi}{\sqrt2}\right)^{1-\chi z}\Upsilon(z), \quad\chi\in\left\{\gamma\frac{2}{\gamma}\right\}.\tag{118}\] Iterating the shift equation 118 , we obtain that, for \(z\in\mathbb{C}\) and \(M\in\mathbb{N}\), \[\label{eq:finite-l-product} \prod_{j=0}^{M-1}l\!\left(z+j\frac{\gamma^2}{2}\right) = \left(\frac{\gamma}{\sqrt2}\right)^{M(2z-1)+\frac{\gamma^2}{2}M(M-1)} \frac{ \Upsilon\!\left(\frac{2z}{\gamma}+M\gamma\right) }{ \Upsilon\!\left(\frac{2z}{\gamma}\right) }\tag{119}\]

Iterating 116 first \(s_1\) times, then on the truncated chain of length \(r-1\), and so on, yields \[\label{eq:iterated-recurrence} \mathcal{I}_{\mathbf{s}}\!\left(\frac{\kappa\gamma}{2},\gamma^2;B_1^+,B_1^-,\ldots,B_r^+,B_r^-\right) = \prod_{\ell=1}^{r}\;\prod_{j=0}^{s_{\ell}-s_{\ell-1}-1}\mathcal{R}_{\ell,j}\,,\tag{120}\] where \(s_0:=0\), and \(\mathcal{R}_{\ell,j}\) is obtained from the recurrence factor by replacing \[r\mapsto r-\ell+1,\qquad \kappa\mapsto \kappa+\gamma(s_{\ell-1}+j),\qquad s_i\mapsto s_i-s_{\ell-1}-j,\] and \[B_i^\pm \mapsto \frac{\gamma}{2}\langle e_{\ell+i-1},\alpha_2\pm m_2\rangle.\] As in the previous step, we have \(\mathcal{R}_{\ell,j}^2=\mathcal{R}_{\ell,j}^+\mathcal{R}_{\ell,j}^-,\) with \[\label{eq:Rljpm} \begin{align} \mathcal{R}_{\ell,j}^{\pm} &= l\!\left(\frac{(s_\ell-s_{\ell-1}-j)\gamma^2}{2}\right)\, l\!\left(1+\frac{\gamma(\kappa+\gamma(s_{\ell-1}+j))}{2}\right)\, l\!\left(\frac{\gamma^2}{2}\right)^{-(r-\ell+1)} \\ &\quad\times \prod_{k=1}^{r-\ell} \frac{ l\!\left( k+\frac{\gamma}{2}\langle h_{\ell+k}-h_\ell,\alpha_2\pm m_2\rangle -(k-1)\frac{\gamma^2}{2} +\frac{(s_{\ell-1}+j)\gamma^2}{2} \right) }{ l\!\left( k+1+\frac{\gamma}{2}\langle h_{\ell+k}-h_\ell,\alpha_2\pm m_2\rangle +\frac{\gamma^2}{2}(s_{\ell+k-1}-s_{\ell+k}-k) \right) } \\ &\quad\times \frac{ l\!\left( r-\ell+1+\frac{\gamma}{2}\langle h_{r+1}-h_\ell,\alpha_2\pm m_2\rangle -(r-\ell)\frac{\gamma^2}{2} +\frac{(s_{\ell-1}+j)\gamma^2}{2} \right) }{ l\!\left( r-\ell+2+\frac{\kappa\gamma}{2} +\frac{\gamma}{2}\langle h_{r+1}-h_\ell,\alpha_2\pm m_2\rangle +\frac{\gamma^2}{2}(s_r-r+\ell-1) \right) }. \end{align}\tag{121}\] Hence \[\label{eq:factorized-square} \left( \mathcal{I}_{\mathbf{s}}\!\left(\frac{\kappa\gamma}{2},\gamma^2;B_1^+,B_1^-,\ldots,B_r^+,B_r^-\right) \right)^2 = \mathcal{R}_{\mathbf{s}}^{+,\mathrm{tot}}\, \mathcal{R}_{\mathbf{s}}^{-,\mathrm{tot}},\tag{122}\] where \(\mathcal{R}_{\mathbf{s}}^{\pm,\mathrm{tot}}:=\prod_{\ell=1}^{r}\;\prod_{j=0}^{s_\ell-s_{\ell-1}-1}\mathcal{R}_{\ell,j}^{\pm}.\)

Applying 119 , using the relation \(h_{\ell+k}-h_\ell=e_\ell+\cdots+e_{\ell+k-1}\) and the neutrality \[2Q-\kappa\omega_r-(\alpha_2\pm m_2)-(\alpha_3\pm m_3)= \gamma\sum_{i=1}^r s_i e_i,\] one checks that

\[\label{eq:Rplus-total-final} \begin{align} \mathcal{R}_{\mathbf{s}}^{\pm,\mathrm{tot}} &= \left[ l\!\left(\frac{\gamma^2}{2}\right) \left(\frac{\gamma}{\sqrt2}\right)^{2-\gamma^2} \right]^{ \frac{2}{\gamma}\langle 2Q-\kappa\omega_r-(\alpha_2\pm m_2)-(\alpha_3\pm m_3),\rho\rangle} \Upsilon(\gamma)^r\Upsilon(\kappa) \\ &\quad\times \frac{ \displaystyle\prod_{e>0} \Upsilon\!\bigl(\langle Q-(\alpha_2\pm m_2),e\rangle\bigr)\, \Upsilon\!\bigl(\langle Q-(\alpha_3\pm m_3),e\rangle\bigr) }{ \displaystyle\prod_{i,j=1}^{r+1} \Upsilon\!\left( \frac{\kappa}{r+1} +\langle \alpha_2\pm m_2-Q,h_i\rangle +\langle \alpha_3\pm m_3-Q,h_j\rangle \right) }. \end{align}\tag{123}\] Finally, using the conditions \(m_1=0\) and \(m_1+m_2+m_3=0\), we complete the proof.

References↩︎